Given a positive integer $N$, prove that there are infinitely many primes $p$ for which there are at least $N$ solutions modulo $p$ to the equation $x^x\equiv1\pmod p$.
A few trivial observations. Clearly the order of $x$ divides $x$. Furthermore, if $x=g^k$, where $g$ is a primitive root, then $\left(g^k\right)^{(g^k)}\equiv1\pmod p$, so $kg^k\equiv-1\pmod p$.