  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.
