Yep we might think the 22nd amendment would preclude Obama from running for president again.
But, if we parse the language, the amendment reads that a person can only be “elected” twice.
Supposed we had a vice-presidential vacancy, and Obama were nominated to be vice-president under the 25th amendment. Then the president resigns or dies in office. Would VP Obama then freely move up to become president again? Could something like that ever happen???
If it ever does, I would assume at that point that he is El Presidente for Life.