Uncaught exception within uncaught exception handler

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?

Well, you have to catch them yourself. The documentation of uncaughtException() linked by you states this clearly:

Any exception thrown by this method will be ignored by the Java Virtual Machine.

But you can only plan so far. So you handle the exceptions of your exception handler but who handles that code? It's the same with logging a failed logging event. Exception handlers all the way down...