Let's skip to the end.  12 bounces.  Surprisingly this only took 25 or so seconds longer than 3 bounces, but there is no reflection going on, so don't expect that in all cases.  Because the difference between 3 and 12 bounces appears to be so small, I suspect that internally the renderer is culling the number of rays calculated per bounce, meaning that with each bounce iteration, a percentage of rays is "thrown away" and calculations are not done for them.  This would account for the **worsening** of quality as more bounces are calculated (rather the opposite of what you'd expect).  This is still with low quality settings though (25%).

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