For a permutation p, let Sn(p) be the number of permutations on n letters avoiding p. Stanley and Wilf conjectured that, for each permutation p, Sn(p)1/n tends to a finite limit L(p). Marcus and Tardos proved the Stanley-Wilf conjecture by a remarkably simple argument. Backed by numerical evidence, it has been conjectured by various researchers over the years that L(p) is on the order of k2 for every permutation p on k letters. We disprove this conjecture, showing that L(p) is exponential in a power of k for almost all permutations p on k letters.