Billiards is one of the most-studied dynamical systems, modeling the behavior of a point particle bouncing around some space. If the space is a plane region bounded by an algebraic curve, then we may use techniques from algebraic geometry to study its billiards map. We explain how to view billiards as a complex algebraic correspondence, and we prove upper and lower bounds on the dynamical degree, the growth rate of the degrees of the iterates, in terms of the degree of the boundary curve. These degree growth rates are studied in mathematical physics, broadly speaking, as a way to identify integrable (exactly solvable) physical models. In our setting, this theory gives us an upper bound on the entropy, or chaos, of billiards in curves.