When performing modular power towers e.g. $a_0^{a_1^{a_2^{.^{.^.}}}}\bmod n$, Euler's totient theorem and it's generalization reduces the problem to computing
$$n,\varphi(n),\varphi(\varphi(n)),\varphi(\varphi(\varphi(n))),\dots$$
until the result divides $a_k$ or the top of the power tower is reached. Suppose that the power tower is tall enough that the top is never reached, and that $a_k$ is never divisible by the $k$th term of this sequence. How many applications of Euler's totient function will it take for $n$ to reach $1$?
Suppose $n$ is a multiple of $2$. Then $\varphi(n)\le n/2$. Otherwise, $n$ is a multiple of an odd prime $p$ and $\varphi(n)$ is a multiple of $p-1$, which is a multiple of $2$, meaning $\varphi(\varphi(n))\le n/2$.
So in the worst case this sequence is $2\log_2(n)\in\mathcal O(\log n)$ long.
Are there better asymptotics on how many applications of Euler's totient theorem it takes to reach $1$? Perhaps a lower bound too?
After the initial step we have $\varphi(x)\le x/2$ since we will always have a factor of $2$. Either $n-1$ is a power of $2$ or the product of a power of $2$ and odd primes, but by the multiplicative identity these odd primes will contribute at least one factor of $2$ on each iteration. This thus proves the tighter upper bound of $\log_2(n)+1$.