The results on Chebychev's bias is a great example. Chebyshev in 1853 noted that primes congruent to 3 modulo 4 seem to predominate over those congruent to 1. This is called Chebychev's bias. Littlewood proved in 1914 that this bias exists and also that it gets violated infinitely often, if you go far enough. The distance to be covered for the violation seemed to grow rapidly with each step.
Rubinstein and Sarnak proved in 1994 that the violations have nonzero density. Here the density means the "logarithmic density", as defined in that paper.
This was a quite interesting result guided at each step from considerations of experimental mathematics and computation. For more information have a look at the Rubinstein-Sarnak paper.