I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the topological entropy or Lyapunov exponent of the system.
Although there is no solution for the general three body problem there are many related results, especially for the restricted case. Here is a study of LE's in restricted 3BP. And here is a classification scheme for the mean motion resonance of the restricted 3BP including using the maximal LE for characterizing the resonances. These will point you to many related results. As a historical aside, Gutzwiller, in his book says that the ancient Greeks knew there were 3 different lunar months (depending on how you measure them in they sky): the Sidereal month (27.32166 days), the anomalistic month (27.55455 days) and the nodical month (27.21222 days). These result from 3 body interactions (earth, moon, sun). The Greeks were apparently able to observe these to an accuracy of 1 second per month. Gutzwiller calls these the first precision scientific measurements in history. Every time I think about this I am astonished; it must have taken immense effort over centuries to compile the needed data.
 Martin C. Gutzwiller Chaos in Classical and Quantum systems.