The first two Vice-Presidents became the second (Adams) and third (Jefferson) Presidents. Those elections, in 1796, when Adams was stuck with a VP of another party, and in 1800, when Jefferson and Aaron Burr had the same number of electoral votes, led to the passage of the 12th Amendment to the Constitution (1804), after which electors voted separately for President and Vice-President, rather than casting two votes for President, superseding a portion of Article II, section 1 of the Constitution.
including four after a President was assassinated, and four after a President died of natural causes.
As a result, some Presidents had no Vice-President for all or part of their time of service. This was rectified by the passage of 25th Amendment (1967) which established a procedure for filling a vacancy in the office of the Vice President, among other issues. This got utilized a few times in the decade after its passage.
Richard Nixon was re-elected President in 1972. His VP, Spiro Agnew resigned over improprieties in 1973, and Congress confirmed Gerald Ford as VP. Then Ford became President in 1974 as a result of Nixon’s resignation over Watergate. Congress then confirmed Nelson Rockefeller to be Ford’s VP.
What of the VPs who never became President? The first of these, Burr, is probably best known for killing Alexander Hamilton in a duel. George Clinton and John C. Calhoun served under two different Presidents. Elbridge Gerry’s behavior in the state of Massachusetts helped create the word gerrymander.
But mostly Veeps are known for the disparaging things they themselves have said about their office, such as these; the John Nance Garner, usually cleaned up to use the word ‘spit’, is the most infamous. It is generally agreed, though, that the VPs in the latter part of the 20th Century and beyond have had far more responsibilities than their predecessors.
Someone came up with a BINGO game so that one could learn the Veeps. I should print this out; I must admit that some of those late 19th-century dudes escape my memory.