In Order of the Phoenix, Fawkes literally swallows Voldemort's Killing Curse, bursts into flames, and gets reborn.

So couldn't he theoretically keep doing this forever? Is there any canon limit to how many times a phoenix can be reborn, or is the only problem that he'd be a tiny chick after each hit?

And theoretically, couldn't Dumbledore just hold baby Fawkes in his hand and use him as a shield against Avada Kedavra?