Three Time Nominee

Generally speaking, politicians only win the nomination for president one or two times.  Parties normally only nominate the politician for a 2nd time if he is the incumbent.  To take examples from recent years, Jimmy Carter, Ronald Reagan, George Bush, Bill Clinton, George W Bush, Barack Obama, and Donald Trump were each nominated for 2nd terms while Walter Mondale, Michael Dukakis, Bob Dole, Al Gore, John Kerry, John McCain, Mitt Romney, and Hillary Clinton were not renominated.  The winners get a second shot while the losers are usually tossed aside.  Of course, there are exceptions.  Sometimes the loser is given a second chance.  For instance, Adlai Stevenson lost to Eisenhower in 1952, but was nominated again in 1956; he lost again.  In a couple of cases, there have been three-time losers: both Henry Clay & William Jennings Bryan were nominees in 3 separate elections.

Looking only at men who served as president, how many of them were nominated on three separate occasions?

Thomas Jefferson was the Democratic-Republican nominee for president in 1796, 1800, and 1804.  He came in second place in 1796, which secured him the vice presidency.  He won the presidency in 1800 and was nominated for re-election in 1804.

Andrew Jackson was one of four Democratic-Republican nominees in 1824.  Though he won the plurality of electoral votes and popular vote, he lost to John Quincy Adams in the contingent election.  He was the Democratic nominee (the Democratic-Republican party had split into factions after the 1824 debacle) in both 1828 and 1832.

Martin Van Buren was Andrew Jackson's chosen successor.  He won in 1836 and was renominated in 1840.  Though he lost in 1840, he became the Free-Soil Party candidate in 1848, marking his 3rd time running in the general election.

Grover Cleveland ran for office in 1884 and won.  However, when he ran for re-election in 1888, he lost to Benjamin Harrison.  He secured the Democratic nomination again in 1892 and won.  He was the first president to serve non-consecutive terms.

Richard Nixon ran for president in 1960.  He lost to JFK.  However, he managed to win the nomination again in 1968 and went on to win the presidency.  He won re-election in 1972, though he didn't finish his term.

Donald Trump duplicated Grover Cleveland's feat.  He won in 2016, lost his 2020 re-election bid, but then came back to win in 2024.

Of this group, Jefferson, Jackson, Cleveland, and Trump were nominees in three consecutive elections.