The St. Petersburg game is a probabilistic thought experiment. It describes a game which seems to have infinite expected value, but which no reasonable person could be expected to pay much to play. Previous empirical work has centered around trying to find the most likely payoff that would result from playing the game n times. In this paper, we extend this work to the distribution of all possible values which could result from this experiment. We use this distribution--with a surprising fractal-like pattern--to examine the unlikely nature of the most famous experiment on this game, the results of the Compte de Buffon's playing the game 2048 times.