Assume the Carmichael's Totient Function Conjecture.

Consider the totient chain

$$n_0=\phi(3^2)\rightarrow n_1=\phi((\phi^{-1}(n_0))^2)\rightarrow n_2=\phi((\phi^{-1}(n_1))^2)\rightarrow\dots$$

where $\phi^{-1}(\phi(x))$ is the smallest second integer $y\neq x$ such that $\phi(x)=\phi(y)$.

(1) Does this chain grow doubly exponentially? (Shown below by Greg Martin)

(2) At every $i$ is there a prime $p_{i+1}$ with $p_{i+1}|n_{i+1}$ and $gcd(p_{i+1},\prod_{j=1}^in_j)=1$? What is the size of this prime?

What if $3^2$ is replaced by $a^k$ where $a,k+1\geq3$ and exponent in $\phi$ is $k$?