22nd Amendment. The most a President can hypothetically serve is a day less than ten years – lifetime; no issues regarding intervening presidencies make a difference. (Subject to an interesting semantic argument, addressed below.)
A President can be elected to office and serve out his term, and be re-elected. After the second term expires, he is not eligible to serve another term.
If a Vice-President succeeds as President, he may serve out the term to which he was elected Vice-President, and then run for office and be elected in his own right. If he served as President for less than two years prior to running for election as President in his own right, he is entitled to run yet again (E.g., LBJ succeeded in 1963, ran in 1964, was eligible to run in 1968). But if he served as President for two years or more of the term to which he was elected VP, he can only run once in his own right. (E.g., Gerald Ford succeeded in 1974, and could have been elected when he ran in 1976, but if he had been, he would have debarred from running in 1980 – but see below.)
Since Pres. Reagan’s death, there is only one living man precluded from running for President by the 22nd Amendment – W.J. Clinton. Messrs. Ford, Carter, and Bush pere are all eligible to serve another term.
Oddball semantic argument: The wording of the 22nd Amendment addresses being elected as President – and by extension based on the 12th Amendment, being elected Vice-President, since no man can be chosen VP who cannot become President. However, the 25th Amendment generates a new way of becoming President – through nomination and confirmation as Vice-President to fill a vacancy, and then succeeding to the Presidency. This is precisely what happened to President Ford – he was named to replace Spiro Agnew as VP, and then succeeded when Pres. Nixon resigned.
Since the 22nd Amendment bars the election of a President who has served two full terms, or one term and over half of another, it can be construed not to prevent such a person being nominated and confirmed by a non-elective process to fill a vacancy, and then succeeding to the Presidency. And we have had in the past several ex-Presidents who were regarded as elder statesmen, who might be perfect candidates to fill the VP vacancy in time of crisis. (I trust it’s clear that Mr. Clinton is not one of them at present.) But, to give an example, if Teddy Roosevelt had survived into the 1920s and the 25th Amendment had been in place then, would anyone have felt it out of place for him to have been named Coolidge’s VP when Harding died?