Let $p$ be a prime. For how many elements $x$ of $\{0,1,\dotsc,p-1\}$ can it be the case that $$2^{2^{2^{2^x}}} = x \mod p?$$

In particular, can you find a simple proof (or, even better, several simple proofs!) of the fact that this can happen only for $< \epsilon\cdot p$ elements $x$ of $\{0,1,\dotsc,p-1\}$?

(Assume, if needed, that $2$ is a primitive root of $\mathbb{Z}/p\mathbb{Z}$.)