The most likely scenario plays out something like this:
The current President dies / resigns / is removed from office after the primary season near the end of his 8 years in office. His VP, who, for some reason, had not sought the nomination (or was defeated for it) assumes the Presidency, proves to be a popular figure, but cannot run for re-election at this point. The candidate of his party is defeated, but, having proved popular in his 10-ish months in office, is nominated four years later and defeats the now-incumbent President.
Although that second campaign would be quite interesting, with both candidates being referred to as "Mr. President," etc.
As others have mentioned, there's always the possibility of a Ford-like scenario (with Nixon resigning in 1975)...the incumbent President, having served less than 2 years in office, is defeated for re-election narrowly, is re-nominated 4 years later and wins that time around. It's rather unlikely that the party involved would give the guy a second chance (especially if it's a Republican), though it is possible.
It's unlikely that a regularly elected President would be renominated, for the reasons stated above---there's an expectation that the President should leave politics, why give a loser a second chance, and, even if he were re-elected, he'd be a lame duck. It's a possibility, though, if the former President in question were extremely popular amongst members of his own party, and his successor had proven to be a disaster (imagine a younger Reagan losing to Gary Hart in 1984, followed by a Monkey Business-filled Hart Presidency, or the 2000 election occurring in 2004 [an extremely narrow loss by Gore for his re-election]).
A more bizarre scenario is the "former President is picked for VP, then becomes President again after a death/resignation." This quite nearly did happen in real life, with Ford. Ford was nearly chosen as Reagan's VP (the deal fell apart at the last moment). Imagine that had happened, and Reagan had been assassinated by Hinckley in 1981. Ford would then have become President again, and, interestingly enough, could have run for re-election in 1984 (Ford, having never been elected President in the first place, would not have been affected by the 22nd Amendment, and could have served over 10 years in office).
The sequence of events that would cause a repeat of that scenario are quite unlikely, however (President dying / resigning, VP loses re-election, VP so un-notable to be considered for the VP slot again, wins election as VP, next President dies/resigns).
Likelihood of any of these happening within our lifetimes? Less than 1 in 10, at best.
Interesting that you bring up the possibility of Ford serving for more than 10 years. This would seem to be a real loophole in the Constitution, actually.
One way that a person could serve limitless numbers of terms would be to be elected President, serve one term, then instead of running for reelection, run for Vice President in the next election instead. If the ticket wins, then the new President resigns on day one and the VP and former President becomes President again. They could just keep repeating this trick every single election with a different candidate on the top of the ticket each time.
