The chaos game is a simple random iterative procedure that generates the attractor of an iterated function system (IFS). This talk will address the question: given a measure driving chaos game, how long does it take for the orbit of a point to reach a certain density inside the attractor? We show under mild assumptions on the IFS and the measure that the rate of growth of this cover time is determined by the Minkowski dimension of the measure. The results will be illustrated with examples, particularly in finding the measure which achieves the fastest possible rate of convergence. Based on joint work with Balázs Bárány and Natalia Jurga.