One could define super Wilson primes as primes $p$ for which there exists a natural number $k$ such that $(p-1)!+1=p^k$. One easily checks that 2, 3, and 5 are such super Wilson primes, where only 5 actually is a Wilson prime. The only other known Wilson primes are 13 and 563, but which are not "super". One easily checks that if $p\ge 5$ is a super Wilson prime, then $k$ must be even.
Question: Are there more such super Wilson primes? Or not?