The iteration $$x_1=r$$ $$x_{n+1}=r^{x_n}$$ with $r=e^{e^{-1}}$ tends to $e$.
What is the smallest index $n$ such that $|x_n-e|<\epsilon$ ?
For small $\epsilon$, it seems that the smallest such index is approximately $\large \color\red{\frac{2e}{\epsilon}}$. Is this true, and if yes, how can it be proven ?