My main question is similar to the title:

Are there any primes $p$ such that $p^3$ divides $(p-1)!+1$?

It is hard to find all $p$ such that $p^2$ divides $(p-1)!+1$ (Wilson primes).

So, in my opinion, there might be no primes $p$ such that $p^3$ divides $(p-1)!+1$. How can I prove this? If it is incorrect, how can I find a prime that satisfies the condition?

Let me know if this question is appropriate or not. If it is inappropriate, I will delete it immediately.