Answer
There are infinitely many primes.
Work Step by Step
Assume there are finitely many primes, call them $p_1,p_2,...,p_k$, where they are sorted in order of size. Then, consider $Q_{p_k}={p_k}!+1$. Clearly, this number is not divisible by any of the primes as they all divide the first part of the RHS and so the entire RHS is 1 mod $p$. But then, $Q_{p_k}$ is prime, but this is a contradiction as $p_k$ is the largest prime. Therefore, there are infinitely many primes.
