Interestingly enough, after 4-6 bounces the light contribution is fairly uniform. It appears that doing more than 6 bounces on this scene wouldn't affect the image quality. A neat hack would be to compute an ambient term from the average light value returned by the 7th-8th bounce, terminate all the rays at that point, and use the computed average value as an ambient term. That might not be the case for other scenes however.
1 bounce.
2 bounces.
3 bounces.
4 bounces.
5 bounces.
6 bounces.
7 bounces.
8 bounces.