We show that the complexity of the billiard in a typical polygon grows cubically and the number of saddle connections grows quadratically along certain subsequences. It is known that the set of points whose first n-bounces hits the same sequence of sides as the orbit of an aperiodic phase point z converges to z. We establishe a polynomial lower bound estimate on this convergence rate for almost every z. This yields an upper bound on the upper metric complexity and upper slow entropy of polygonal billiards. We also prove significant deviations from the expected convergence behavior. Finally we extend these results to higher dimensions as well as to arbitrary invariant measures.
