The Bunimovich Stadium is one example of a Dynamical Billiards table that exhibits chaotic behavior even with only concave scatterings. Trajectories in the Bunimovich stadium exhibit (eventual) exponential divergence over time. In addition, almost any trajectory in a Bunimovich stadium will eventually touch cover any point of the stadium. This is in sharp contrast to trajectories in elliptical billiards: any one trajectory in elliptical billiards will always leave a chunk of the table unexplored.
Inspiration for this post came from Baez’s post on the stadium.