259

This number is a composite.

259 = 7 x 37 and there are 7 primes between 7 and 37. [Beldon]

The product of first 259 even numbers plus 1, i.e., 2^259*259!+1 is the only non-titanic prime of form 2^n*n!+1 where n is an integer. [Loungrides]