TITLE: Is there any nonconstant function that grows (at infinity) slower than all iterations of the (natural) logarithm?
QUESTION [15 upvotes]: Is there any nonconstant function that grows at infinity slower than all iterations of the (natural) logarithm?

REPLY [24 votes]: Let $\text{LogNestMonster}_i(n) = nest(log, i)(n) = \overbrace{log(log(...(log}^\text{i times}(n))...))$.
Many functions grow slower than $\text{LogNestMonster}_i$, no matter how large the $i$ you picked is:

The inverse Ackermann function $\alpha$ grows slower than all $\text{LogNestMonster}$s. It (roughly) tells you how far in the sequence $1+1$, $2 \cdot 2$, $3^3$, $4\uparrow\uparrow4$, $5\uparrow\uparrow\uparrow5$, $...$ you have to go before exceeding $n$.
The iterated logarithm $log^*$ also grows slower than all $\text{LogNestMonster}$s. It counts how many times you have to apply log to $n$ before the result is less than $1$.
Also, consider the function $\frac{1}{n}$. It is not constant and is clearly asymptotically less than all $\text{LogNestMonster}$s, although I'm guessing it's not exactly what you had in mind since $\frac{1}{n}$ is $O(1)$ despite not being $\Theta(1)$.