I have tried to find a positive integer $x$ such that $e^{{e}^{{e}^{x}}}=2^{n}$, for $n=2335$. I have got $x$ close to $2$ as shown here, then my question in general is :
Question: Is there any paire of positive integers $ (x,n)$ for which $$e^{{e}^{{e}^{\cdots x}}}=2^{n}?$$