Depending on whether the annuity is fixed or variable, immediate annuities can have various drawbacks ranging from loss of purchasing power from inflation (with a fixed annuity), or high fees (with a variable annuity).
For example, a fixed annuity guarantees you a set payment for a long period of time - possibly the rest of your life. But you might live longer than you think. Those payments you started getting when you first retired won't change at all, and they might seem pretty measly after 40 years of inflation.
A variable immediate annuity might keep your payments from being eroded by inflation - but then again they might not. Payments bounce up or down each month depending on the performance of your underlying investments, which can make budgeting tricky. And payments can drop substantially over the short term if the markets fall.