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(Did not read yet, pardon my surmises.)
It sounds as if one of the continuities which does not provide a death is the Batman Beyond storyline (Though the DCAU joker appears?). In that storyline Bruce puts away the mantle and passes it along to someone else. So *A* Batman continues, but it isn't Bruce.
If, in all other lives, Bruce lives as batman, dies as batman, and is reborn as batman because he cannot stop being batman, is Beyond the final life? The life where he has done enough and can move on? Because he was able to stop being batman and pass the mantle on to Terry?
Wouldn't the fact that he is reincarnated as himself make it an example of a stable time loop?
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How well does it match the trope?