This question may sound a little silly ;)
How would approach the possibility of an uncaught exception within the UncaughtExceptionHandler?
Very interesting question, I haven't thought about this before.
The approch seems to be that either set your own uncaughtExceptionHandler on your thread or it will get passed to JAVAs defaultUncaughtExceptionHandler,
I made some quick googeling and found a similar case for getting crash data from android. I think you may be intresing in this link How do I obtain crash-data from my Android application?