Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here.
Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ for $n\in\mathbb{N}$, where $\zeta$ is Riemann zeta function. $\textbf{Question}:$ What we can say about the convergence of $(a_n)_{n\in\mathbb{N}}$ ? If this sequence has a limit what we can say about it ? Is it a rational number ?
I tried to do some numerical calculations to see how it behaves for large $n$ and it looks like this limit is equal $1$, but I don't have any idea how to prove it.