For each odd $n>2019$, either $n$ or $n!-n$ works. To see why, let $n!+1=p$ be a prime. Then by a variant of Wilson's Theorem, $$n!(p-1-n)!\equiv (-1)^{n-1}\pmod p\implies (n!-n)! = (p-1-n)!\equiv -1\pmod p$$thus $p\mid (n!-n)!+1$ so $n!-n$ works.