Actually, this is not precisely my question. If $a(x)$ is the inverse ackermann function, then obviously $a(a(x))$ grows slower than $a(x)$, as does $\log(a(x))$, and so on. But is there a function f that goes to infinity and cannot be asymptotically bounded below (when x goes to infinity) from any finite composition of logarithms, inverse Ackermann functions, and arithmetic operations?
(Equivalently, is there a "well-behaved" function that cannot be bounded above by a finite composition of Ackermann functions, exponentials, arithmetic operations, etc?)
Edit: I mean functions that are defined for all (positive) reals, not just integers.
Edit 2: Okay, infinitely differentiable functions.