Relaxation of the $\pi^{\pi^{\pi^{\pi}}}$ problem

Question

My friend showed me this fascinating problem:
Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?.

Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger.

It turns out that we don't know whether $\Large \pi^{\pi^{\pi^{\pi}}}$ is an integer. This is because a) there are very few theorems about combinations of transcendentals and b) the number is too large to reason about computationally.

So I was thinking about a relaxation of the problem. Let $f(a)=\Large a^{a^{a^a}}$. Can we find a rational $q \neq 0$ such that $f(q*\pi)$ or $f(q+\pi)$ is rational?

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I have tested all $q=p/r$, such that $1 \leq p,r \leq 100,000$ and $q*\pi \leq 2$ and did not find any integer $f(q*\pi)$. The most interesting was $q=18868/37909$, which gave $f(q*\pi) \approx 3$.

Similarly I have tested all $q=p/r$, such that $1 \leq p,r \leq 100,000$ and $|\pi-q| \leq 2$ and did not find any integer $f(\pi-q)$. The most interesting was $q=85186/61765$, which gave $f(\pi-q) \approx 14$.