Spoiler for

If p>3 is prime, consider the set of integers between 2 and p-2 inclusive. Each number in this set can be uniquely paired with another number in this set so that their product is congruent with 1 modulo p. Hence (p-2)! = 1 modulo p and consequently (p-1)! = -1 modulo p, so (p-1)!+1 is composite as it divides p. Let n=p-1 and n!+1 is composite, and since there are infinitely many primes there are infinitely many such n.

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