George Washington served two terms and indicated that he thought that was enough. Whether he meant for himself, or a man of his age or for anybody is unclear (I believe). It became the custom (no more) for Presidents to serve a maximum of two terms. Grover Cleveland was the only President to serve non-consecutive terms. Ulysses Grant wanted to run for a third term but was vehemently oppposed. There was similar opposition to Teddy Roosevelt when he tried (unsuccessfully) to run for what would have been a third term in 1912. In the event his candidacy split the Republican vote and allowed Woodrow Wilson to be elected for the Democrats.
FDR was first elected President in 1932, stood successfully for re-election in '36 and then against much opposition went for a third term in 1940. In '44 he argued that it was the wrong time to have a change of leadership in the middle of a war and got in for a fourth term. Harry Truman inherited his Presidency the following year on Roosevelt's death and went on to serve for a further term.
It was in the background of the recent example of the FDR Presidency, and a continuous 20 year Democratic occupancy of the White House that the Senate passed the 22nd Amendment limiting any President to a maximum of two terms. The amandment was justified most strongly in preserving the balance between the Congress, Executive and Judiciary. The lack of a limit on the length of the Presidency was felt to confer too much power on the Executive.