Does $n^2+1\nmid n!$ hold for infinitely many $n\in\mathbb N$?
OEIS entry A144255 states that H. Iwaniec, Almost primes represented by quadratic polynomials, Invent. math. $47$, $1978$, p. $171$–$188$, proves that there are infinitely many primes or semiprimes of the form $n^2+1$. If $n^2+1$ is a prime, then it doesn't divide $n!$. If it's a semiprime, one of its two prime factors is $\ge\lceil\sqrt{n^2+1}\rceil=n+1$, so again it doesn't divide $n!$. Thus there are indeed infinitely many $n$ such that $n^2+1\nmid n!$.
Whether there are infinitely many primes of the form $n^2+1$ appears to be an open problem.