TL;DR: Calvin Coolidge, James Polk, and Rutheford Hays are possible answers, though since they never ran for another election, we'll never know for sure.
Here's my answer to a very similar question a few years ago, cleaned up:
The list of presidents is pretty short, so we can just go through them one by one.
There were 45 presidents in total. We then eliminate those who ran for second term, as well as those who would have run but either lost renomination (Chester Arthur), were about to lose renomination (Lyndon Johnson), or were expelled from their party (John Tyler). After doing this, we are left with five presidents:
James Polk: During his election in 1844, he promised to run only for one term. As promised, he did not seek re-election. He died of cholera shortly after leaving office.
James Buchanan: Various sources seem to suggest that he "was tired of presidency" and that's why he did not seek re-election in 1860 (when Lincoln won). Alternatively, he may have just been unpopular because his original nomination was a compromise between Southern and Northern Democrats, and by the time his term was up the differences between the two were pretty irreconcilable. Further reading is required to determine whether he just knew that he couldn't get elected or whether he voluntarily decided to pack up and leave. Either way, he remained politically active.
Rutheford Hayes: He pledged to run only for one term and did not seek re-election. Not quite sure what factors went into his one-term decision.
Calvin Coolidge: He finished Warren Harding's term (who died early in the presidency), and did another term. When asked whether he'll run again, he said that he wanted to take a break from the White House because he's tired of it. It's a good idea to note that had he run and won, this would have been his 2.5th term, not 2nd term, which may have played a role in his decision. More on that here : "I do not choose to run".
Harry Truman: His situation was similar to LBJ's, i.e. he was unpopular. Unlike LBJ, he just decided not to run outright. In addition, his re-election would have violated (in spirit, though not in law) the 22nd amendment, which was then just passed. The amendment set the limit of presidential time in office to 2 terms or 10 years. For various procedural reasons it did not apply to Truman, but it would have been impolite to violate it.