 One can take a suitable branch of the logarithm of an entire function that never hits 0, so that this will also be an entire function (according to the Weierstrass factorization theorem).The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times.Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.