Abstract: In a hyperbolic dynamical system (map) T with singularities, the "complexity" of the singularity structure K(n) is the number of smoothness components of the n-th iterate T^n that can meet at a single phase point. Having an upper bound on K(n) plays an important role in many arguments about statistical properties, including Young tower constructions in dispersing billiards. In this talk I show that K(n) is bounded (by 4) for C^{^3}-typical planar dispersing billiards with finite horizon and no corner points. This result has been unpublished since 2006, because it had no applications: it seemed that the much easier linear bound is enough for everything, and the much more interesting multi-dimensional generalization is hopeless. However, it turned out recently that there are people interested in the finite complexity case: new results of Baladi, Carrand, Demers and Korepanov on the measure of maximal entropy require stronger than linear bounds, and my result implies the typicality of their conditions.