(FORTUNE Magazine) – Maurice Raymond "Hank" Greenberg, the chairman of giant insurer American International Group, turns 73 next month and often--too often for his taste--hears questions about his retirement plans. In February, sitting in a big leather chair in the elegant wood-paneled and Chinese-red anteroom of his New York office, he looked sideways at a visitor who was the latest to inquire and airily dismissed thoughts of packing it in: "I feel good," he said, "I like what I do, and I haven't had another job offer."

We will pause here for executive headhunters to straighten in their seats. Greenberg may have been Jack Benny's age for decades, but if anybody thought he could possibly be serious about other jobs, he'd be up to his chinoiserie in bids. In the world of bigtime financial services, he is widely and enthusiastically admired for his management abilities and for his long-term consistency in pulling outstanding performance out of AIG.

Think really long term: This is Greenberg's 31st year as CEO. When he rose to the job in early 1968, AIG was a private company, and before 1969 was out, he'd taken it public. It had a market value then of about $300 million, has mainly grown internally rather than by issuing stock, and today has a market value of $88 billion. Remember, moreover, that this trip occurred in the insurance industry, which is bruisingly competitive, stingy with profits, and of late particularly hazardous for AIG's breed, the "multilines." Those are the sellers of both property-and-casualty (P&C) and life insurance, and they've had difficulty staying alive. All of Travelers and much of Aetna were swallowed up by Sandy Weill; Cigna has meanwhile staggered under P&C problems.

That may suggest just how difficult these big companies have been to manage--and yet here's AIG, whistling along, though it is just about the most complex company in captivity. It operates in 130 countries; juggles more than 300 insurance subsidiaries; and has spent the past decade busily and sometimes controversially buying businesses, such as aircraft leasing, in which it had no expertise.

It had Hank Greenberg, though, who in the face of everything awesome about the assignment kept turning out earnings gains. In the 31 years he's been running AIG, earnings per share have dropped in only one year (1984, a brutal time for P&C); for the entire period they have grown at an annual compounded rate of 19%, which is sensational. The growth rate slowed in the decade to just over 12%. But AIG is a huge company now--it's No. 26 on this year's FORTUNE 500 list--and hard to move along at racehorse speeds.

Which doesn't keep its jockey, Greenberg, from working the whip in every ride. This man, you should understand, can be charming. But he is also tough, demanding, impatient, focused, tireless, tenacious, and just about every other adjective in the thesaurus that suggests drive and determination. Personalities like that don't coexist easily with growth rates down around 12%, and Greenberg in fact burns to get at least 15%. Last year (when AIG's profits were $3.3 billion), he pulled that off, with 15.9%.

That was a true feat, because 1997 was rudely interrupted by the problem called Asia. This is no ordinary region for AIG: It's the fatherland. In a claim that no other FORTUNE 500 company appears able to make, this business was founded in Asia--in Shanghai, in 1919, by a money-short, 27-year-old Californian named Cornelius Van Der Starr, who ran the company for 49 years before lateraling it to a man he'd hired eight years earlier, Greenberg. (Therein, of course, lies an extraordinary fact: In its long history, AIG has had only two CEOs.)

Today, AIG also has enormous insurance operations, both P&C and life, in Asia. In 1997, the company made pretax profits of $1.7 billion in the Far East, more than one-third of the company's total. That is way past what Citicorp earned in the region ($1.2 billion) and more than Coca-Cola made in the Far East and Middle East combined ($1.5 billion).

AIG's big bucket of profits from the Far East last year suggests correctly that the area kept doing its stuff for the parent's bottom line. But it contributed less to growth than in previous years, and the buckling of so many Asian economies raises questions about how AIG will do in the region in the near term and perhaps longer. So does this worry the stock market? It does: In the past few months, AIG stock has swooned periodically because of bad Asian news. In December, Hank Greenberg even held a special fireside chat for analysts to soothe their fears.

But what really reassured the market--and pushed the stock to a succession of new highs --is that AIG, despite Asia, came through with its usual "up" earnings, and especially good ones, to boot. In short, Greenberg proved once again that he has the formula. This time he did it with banner results in the financial-services business he's been building in the past decade and with profits pulled out of the myriad nooks and crannies of AIG--life insurance in Poland, for example, and P&C in the U.K.

Those are just details--just substantiation that Greenberg's got the right stuff. Fact is, many Wall Streeters have given up on really analyzing this company: It is so complex that they think it inscrutable. They just fall back on faith, telling themselves that Greenberg will keep turning out the earnings.

That naturally raises the question of who will turn them out after Greenberg. Maybe he will be like Warren Buffett, who jokes that after his death he expects to be running Berkshire Hathaway by means of seances. But there is another possible answer at AIG, name of Greenberg. Last year the AIG board asked one of its own, Thomas R. Tizzio, now 60, to vacate the president's job at AIG so that the post--enlarged to include the title of chief operating officer--could be given to Hank's second-oldest son, Evan Glenn Greenberg, now 43. (Tizzio, the company's P&C guru, became senior vice chairman.) The younger Greenberg has worked at AIG for 23 years, doing tours of duty in many different quarters and, for a time, heading all Japan operations. Within AIG, there's every thought that Evan will follow Hank as CEO.

Outside the company, the first thought is often "nepotism." Not that family dynasties are unprecedented in large publicly owned corporations: There have been, for example, the Thomas Watsons at IBM, the John McCoys at Banc One, and the Galvins, Robert and Christopher, at Motorola. But in a world of professional managers, those situations are still rare enough to raise eyebrows. Can we imagine, for example, an offspring of Jack Welch's running GE or a kid of John Reed's taking over Citicorp?

At AIG the circumstances are even more intriguing, because Hank Greenberg's oldest son, Jeffrey, now 46, was until 1995 also a senior executive of the company--and another apparent contender to be his father's successor. But he then left, soon after surfacing as head of the risk-capital operation at the U.S.'s largest insurance broker, Marsh & McLennan, which does tons of business with AIG. Neither Hank nor Jeffrey Greenberg cares to explain why the latter left; Hank, in fact, flamed up at FORTUNE for even raising the subject.

Evan Greenberg, on the other hand, recently supplied a small answer: "He just needed to do something different for himself." Perhaps, say some outsiders, he especially needed to do it away from the intense, autocratic, nothing-matters-but-this-company environment to be found at AIG. In any case, Jeffrey may be another Greenberg headed for the top. Within Marsh & McLennan, he is regarded as a smart, analytical executive who has an excellent chance of succeeding CEO A.J.C. "Ian" Smith, 64.

The son who stayed at AIG and became president, Evan, is often described as less intellectual and more "intuitive" than his brother (that seems to mean quicker to form judgments and act on them) and cast more in the mold of his father in being tough, and even, says one insurance CEO, "abrasive." Another executive who has seen Evan in action does not think him "the sharpest tool in the toolbox," but many others think him plenty smart. Know this also: Evan has been a maverick in his life and one of the few people to have had the gumption to assert himself against Hank Greenberg (see box).

Though it is certainly not their favorite subject, both the son and the father recently stepped up to the question of why Evan deserved the president's job. Said Evan Greenberg: "My father is a very disciplined man. He built this company; he loves this company. He will always make the tough decision of what's right for this company. And he wouldn't make this move if he absolutely didn't have confidence in his own mind--maybe more confidence than I have--that I'm the right person for this job." And besides, he said, there's AIG's board and a set of current and retired company executives whose opinions matter, and they seem to concur.

In his answer to this sticky question, Hank Greenberg recalled that AIG has always been a "family-oriented company" that has employed "generations" of relatives. "But," he said, "I would not be governed by that only. I would recommend and choose the best person"--regardless of his name--"that I think would be capable of managing and providing the leadership to this organization." And Evan is that person? "Absolutely," he answers.

To an extent, a family dynasty matches the idiosyncrasies of this company, whose history has left it looking like a public surrey with a private fringe on top. AIG owns a private golf course, Morefar, north of New York City, and it also owns Mt. Mansfield Corp., the proprietor of Stowe, the ski resort in Vermont. More fundamentally, AIG has one of the strangest ownership structures around. That even became an issue this year in the battle between AIG and Cendant over which would get to pay $3 billion to buy American Bankers Insurance, a Miami credit insurer. Cendant won, so to speak, after Greenberg dropped out of a price war (pocketing a $100 million termination fee as solace). During the fight, the two companies traded insults and charges--and one slur thrown by Cendant accused AIG of being controlled by "shadowy offshore companies."

It is true, to parse that charge, that 29% of AIG's stock is controlled by management and that some of the control is exercised through private companies whose character is only fuzzily described in AIG's proxy statements. As the proper adjective, though, "shadowy" deserves some competition from "incestuous."

Much of the control originates with AIG's compensation programs. First, there are about 1,000 AIG people (out of 40,000 employed) who have stock options--nothing particularly unusual about that. Second, about 300 of the 1,000 who are especially valued by the company have been--and may annually continue to be--awarded a weird form of restricted stock, which they will get their hands on if they stay around until they're 65. (This company is a specialist in golden handcuffs.) The restricted stock plan gives these 300 people units of participation in something called Starr International Co. (SICO), which is incorporated in Panama and is one of the "shadowy companies" Cendant had in mind. By way of its ownership of 16.2% of AIG's stock (recently worth more than $14 billion), SICO is the company's largest shareholder. That's a chunk of stock, and it's a big reason why nobody ever talks about taking over AIG.

Third and last, there's another strange something called C.V. Starr & Co., in which about 40 of AIG's really, really valued people hold stock. Starr is several things: a private holding company that owns about 2.4% of AIG's stock, which adds to the takeover protection; a collection of insurance agencies that do business with AIG; and a "you've made it, kid" club that every aspiring AIG executive wants to be invited into. True, to join this club an executive has to pay a sort of initiation fee: the cost of buying the Starr shares he's been allocated. But who cares? Many of AIG's executives remember exactly where they were when they got word that they'd made this fraternity. The next thrill, says Evan Greenberg, is when you're made a Starr director (he's one), and maybe the thrill after that is when you find out you're succeeding Hank Greenberg, who owns 25% of Starr's stock.

As all this suggests, there's more to being a Starr shareholder than just the money. But the money is okay too. Starr's AIG stock keeps rising in value--it's up to $2.1 billion--and brings in some dividends besides. Then there's that insurance-agency business, which gets us to the incestuous angle. Starr, this private company owned by AIG executives, generates policies and premiums for AIG and gets paid for that. Last year AIG made net payments to Starr of $32 million. That's not bottom line for Starr, because it has some expenses to cover. But once they are paid, what's left belongs to those 40 people.

There's only one problem: The Starr shareholders can't get their full stake out until they're 65. That golden handcuff, plus SICO's, tends to chain the highly prized to the company. One member of the corporate establishment remembers trying in the early 1990s to lure Edward E. Matthews, then CFO of AIG and now its chief investment officer and financial-services boss, away from AIG. No way, Matthews is recalled to have said: He'd have to leave $35 million on the table. And that was years ago, when AIG stock was a pip-squeak compared with what it is today.

As for Hank Greenberg, he's got his 25% interest in Starr; a stake in SICO that is worth many (though undisclosed) millions; and, in the owned-outright department, 16.2 million shares of AIG stock recently worth just over $2 billion. If you seek proof that this man is not glued to the job by a need for money, there it is.

Over the decades in which Greenberg has amassed this wealth, and in which 40 men now at AIG (yes, they're all men) have risen to the Starr stratosphere, the ranks of the company's executives have also been thinned by many departures. That's a well-broadcast fact about AIG, and FORTUNE knows it with some specificity because we checked out the fates of all the executives whose pictures were in the 1987 and 1988 annual reports. (In later reports, AIG cut the number of pictures run.) Of the 60 men pictured, 23 are still there; 21 have retired or died; and the remainder, a big 16, left.

Some, probably including a few who saw themselves handicapped in not being named Greenberg, headed for opportunities they thought better; many another insurance company, indeed, has been sharpened up by former AIG executives. But also among those departing have been numerous executives who just couldn't cut the mustard with Hank. "He doesn't suffer fools gladly" is a line that could have been minted at AIG.

As a matter of fact, Greenberg doesn't suffer smart people gladly if they happen to challenge something he holds dear--like the proper operation of his company. A case in point concerns Howard Sosin, who came to AIG in 1987 from Drexel, bringing with him a sophisticated financial-products operation that sold derivatives and the like and that became the nucleus of AIG's push into financial services. By 1992, Sosin's operation, called AIG Financial Products, was delivering around $172 million in pretax profits.

The trouble, though, was that Sosin wanted to run things in an all-out way that accelerated the recognition of profits (of which Sosin got a rich cut) and that also created more risk than Greenberg, a financial conservative, could abide. A still deeper problem, to get right down to it, was that Sosin and Greenberg were both control freaks, and that was one too many. So in early 1993, Greenberg announced that Sosin would leave because of "a difference in opinion." AIG also took special charges that year that sliced off some of the profits Sosin had been reporting. Greenberg meanwhile put in new management and settled back to running the operation his way, which seems to be clicking: Last year Financial Products had pretax profits of $241 million.

The Greenberg style of managing also includes strong doses of centralization. True, you can't run a far-flung operation like AIG from New York. But you can have budgets and constant vigilance and a workaholic CEO who's not only technically expert in insurance but also unbelievably tuned into detail. Says a Wall Streeter who has long known Greenberg: "There is not a stone that drops in that company that he does not hear."

Greenberg, for example, is a fanatic about AIG's internal auditing, conducted by 100 people who roam the company looking for wrongdoing. Anything they spot is reported directly, in writing, to Greenberg and a few other executives, including CFO Howard Smith. Over time, says Smith, he's learned that he'd better get to the reports immediately or he will have the unwelcome experience of not having read a report, even as Greenberg is on the phone asking him for comment. Recently, Greenberg heard from a visitor that some companies seemed to be "outsourcing" their internal audit (an oxymoron if there ever was one). At this news, Greenberg threw back his hands, rolled his eyes, and in effect asked that his ears be spared such rot.

In another bout of inspection, Greenberg spends perhaps 50 hours every fall in budget meetings with upwards of 25 different operating units. Each unit is armed with a book containing its business plan. Greenberg, says Smith, will unfailingly have read every word of these books and be loaded with questions. The meetings, says Smith with a grin, can be "lively and contentious." And is it permissible for a unit to forecast no gain in earnings? "You'd better have a decent explanation," answers Smith, his grin widening. But then he backtracks and talks about the retreats that AIG determinedly makes from businesses not producing adequate profits--workers compensation in California, for example. In these businesses, forecasts can definitely be down.

Above all, Greenberg expects in these meetings, and in every moment of his days, to hear about innovations and ideas for staying ahead of the curve. He himself took AIG into a huge swerve a dozen years ago, deciding that the P&C business--then booming--was bound to soften (which it's never stopped doing since) and that AIG had better look around for auxiliary sources of profits. From that stretch of soul-searching came AIG's financial-services business: the derivatives operation; a commodity and currency trading company; and International Lease Finance Corp. (ILFC), which has around $14 billion in planes leased out to carriers all over the world.

ILFC was expensive for the day, costing $1.25 billion, and one Wall Streeter remembers "kvetching around about the price," sure that AIG had gone off the deep end. But, he says, Greenberg and Ed Matthews had it figured out that this was an investment, to be viewed as an alternative to buying, say, a bond that paid 7.5% interest. And for sure, Matthews doesn't get a chance to buy many bonds that yield what his financial-services businesses do. Their return on equity last year was north of 20%.

In another part of AIG, today's innovation includes a startup credit-card operation in the Philippines. AIG has long had a co-branded card venture in certain Asian countries with Standard Chartered Bank. But the Philippine push puts AIG on its own with cards and in a business it hopes can really grow. To plug some expertise into the startup, AIG has hired Pei Chia, the former head of Citicorp's consumer business, as a consultant (and has also put him on its board of directors). Chia's presence in the mix is a sign of Greenberg's covering his risk, which he does in every part of his business, in any way possible.

Part of the drill for AIG in Asia is simply stretching its reach. The company's history in China, as the major example, has been stormy: AIG was evicted by World War II; re-entered after the war and was evicted again, this time by the Communists; and then re-entered again in 1992, after many years in which Hank Greenberg, turning on all his persuasive powers, persistently lobbied for operating privileges. AIG has these now in two Chinese cities, Shanghai and Guangzhou (which is near Hong Kong), and Greenberg is still working to enlarge AIG's territory. It helps, of course, that AIG's unwavering practice since 1919 has been to hire local people and load them with responsibility. Today, for example, the head of AIG's worldwide life insurance operations (and an AIG director besides), is Edmund Tse, 60, whom the company hired out of the University of Hong Kong in 1961 and trained as an actuary.

For AIG, in its drive to sell life insurance, the continent of Asia looks like a vast exploitable market, heavily peopled with prospects not yet jaded and picky about insurance (in the way that Americans are) but instead happy to buy policies that feature a big savings component. These are high-margin products and one reason that AIG's life profits have historically grown so fast.

As an absolute opposite, there's AIG's P&C business in the U.S., in which it has historically been a seller to corporate elephants spread around the world. This business is extraordinarily competitive these days, with no turn in the market really visible. So AIG has had to focus creatively on its specialties--environmental risks, for example, and kidnap-and-ransom policies--and also rustle up business in spots it once didn't worry about.

In one instance, Greenberg called up his longtime friend Frank Zarb, chairman of the National Association of Securities Dealers, and asked how AIG might "add value" to Zarb's stable of Nasdaq companies, many of them smaller than would normally appear on AIG's radar screen. It turned out that Nasdaq was just broadening its requirements that its companies have outside directors--and these folk won't sign on, thank you, if they aren't covered by directors' and officers' liability insurance. So AIG has tailored a set of policies that Nasdaq (in return for a fee on business done) is now marketing to its companies. And incidentally, don't think the Greenberg/Zarb connection got Nasdaq any special favors: "I learned a long time ago," says Zarb, "that you have two distinct relationships with Hank: One's friendship, and one's business--and this was business."

Right now, in a P&C thrust that could be important for future profits, AIG is pushing hard to build an auto-insurance business in the U.S. It's a business that has long teased Greenberg: He has been intrigued by its bigness--$112 billion in premiums last year--but repelled by the distribution costs it carries if sold through the normal channel, agents. Direct marketers, though, get around the distribution costs, and in the early 1980s, Greenberg tried to buy a leader in this field, GEICO. Negotiations, however, came to nothing.

By the 1990s, Greenberg simply concluded it was time to move. On one front he made investments in two low-cost auto insurers, and on a second front he cranked up a direct-marketing operation inside AIG. He's using TV ads to build a brand around the AIG name, and Robert Sandler, who heads auto insurance, says the campaign's working. At the least, the business is growing: Last year premiums rose 11% and profits 50% (to $70 million).

Sandler says he's after steady double-digit premium growth. That's tough in a business growing only 5% a year, since you've got to steal revenues from competitors. But Sandler says that's fine: "It makes you a predator, and that's a comfortable position for AIG."

Predator, entrepreneur, martinet, innovator--Hank Greenberg answers to all. Fitness nut, too: He works out most mornings, watches his diet, and skis (says FORTUNE's photographer, an expert skier himself) like a man 50 years younger.

There will someday come a time, nonetheless, when the helm of this giant and complex company will need to pass to a third chief executive--perhaps named Greenberg, perhaps not. In a way, that matter won't be the one of first consequence. What will ride above it is the question of whether anybody but Hank Greenberg can really run AIG. It is a great company now, no question about that. But there is a downside to a greatness that is so centered in one individual. When that person has been an overpowering presence for 30 years and longer, and when he has so completely made a company in his own image, and then, when he's not there--that's when the greatness of what he built will be tested.

REPORTER ASSOCIATE Eileen P. Gunn