Without the 22nd Amendment, which presidents would have run for a third term?

[Simfan34]

It is one of the great ironies of history that the 22nd Amendment, introduced by the Republicans to forestall the possibility of another FDR being elected for four terms, has until now served to scupper the potential ambitions of Republican presidents; that is changing, with Clinton and Obama ending their terms relatively popular.

But I actually don't think any President since would have run for a third term even if they could have. I imagine someone's asked this question before, and I even wonder if I have myself, but I'm curious as to what you think. Here's my breakdown:

Truman-- was able to run again, chose not to
Eisenhower-- Health issues probably would have preempted a bid for a third term, also would have probably simply been disinterested in another term; from what I've read, in 1956 he already was happy enough to leave most of the campaigning to Nixon.
Kennedy--Assassinated
LBJ-- Declined reelection
Nixon-- Resigned
Ford-- Lost reelection
Carter-- Lost reelection
Reagan-- Even if you accept the official position that his Alzheimers was undiagnosed until after he left office, there were already clear health issues recognized semi-publicly which would have ruled out another term, if not by his administration's own admission, then at least by public opinion
Bush the Elder-- Lost reelection
Clinton-- Admittedly, he left office popular, but this seems to me to have at least partly been a sort of early post-presidency popularity boost. His absence from much of the 2000 campaign testifies to the degree to which many, even in his own party, saw him as a liability rather than an asset. Heck, going by how they talk about Bill today it's clear many in Clinton-land still see him as a liability. So I suspect that any desire to run again would have been gently scuppered if for no the reason than to prevent him from destroying Hillary's chances at advancing her own political career.
Bush the Younger-- Obviously far too unpopular to even think about it. Probably would have been primaried had he tried, actually.
Obama-- From his public comments he has zero desire to stay in office for even a day after January 20, 2017-- as evidenced by interviews and especially his speech to the African Union. However I fully concede that these comments are made in a world where a third term is an impossibility and thus probably hasn't been seriously considered. He would have probably felt at least a little obligated to step aside for Hillary, especially considering 2008, but maybe explicit demand for a third Obama term would have pushed Biden to run on a continuity platform. Still, I think Obama would stay out. But he could have been urged to make a comeback in 2020 after Hillary's approvals tank (remember: they're already in the tank)

Ultimately, I believe, the two term custom would have retained considerable weight in presidential decisions, even after FDR broke the tradition-- it would have (and is) probably been rationalized as an exception given the dual exceptional circumstances of the Great Depression. There'd be something of a standard requiring presidents to "prove" that circumstances justified their bid for a third term. If nothing else, the fact that of all the Presidents who could have done so, just two sought third terms.

But what do you think?