Iterated Euler's totient function

Note that $\phi(n)$ is even (for $n\ge3$), and if $n$ is even then $\phi(n)\le n/2$. This immediately gives you Pillai's logarithmic upper bound.

Erdős et al. say this in On the Normal Behavior of the Iterates Of some Arithmetic Functions:

[...] it is easy to see that the set of numbers of the form $k(n)/ \log n$ is dense in $[1/ \log 3,1/ \log 2]$. What is still in doubt about $k(n)$ is its average and normal behavior. We conjecture that there is some constant $\alpha$ such that $k(n) \sim \alpha \log n$ on a set of asymptotic density $1$.

Here, $k(n)$ is what the OP calls $\Phi(n)$.

The original paper was published in 1990. Perhaps it is still the state of the art.