We show that for infinitely many prime numbers p there are at least \log\log p / \log\log\log p distinct residue classes modulo p that are not congruent to n! for any integer n.
BANKS, WILLIAM D.; LUCA, FLORIAN; SHPARLINSKI, IGOR E.; and STICHTENOTH, HENNING (2005) "On the Value Set of n! Modulo a Prime," Turkish Journal of Mathematics: Vol. 29: No. 2, Article 6. Available at: https://journals.tubitak.gov.tr/math/vol29/iss2/6