Factorial primes are primes of the form $n!\pm1$. (In this application I'm interested specifically in $n!+1$ but any answer is likely to apply to both forms.) It seems hard to prove that there are infinitely many primes of this form, though Caldwell & Gallot were courageous enough to conjecture that there are infinitely many and to give a conjectured density ($e^\gamma\log x$ below $x$).
I'm looking at the opposite direction: how many composites are there of the form $n!\pm1$? It seems 'obvious' that the fraction of numbers of this form which are composite is 1, but I cannot even prove that there are infinitely many composites of this form.
Has this been proved? (Perhaps there's even a proof easy enough to relate here?) Or on the other hand, is it known to be open?