Teddy Roosevelt, with his Bull Moose Part in 1912 was probably the closest - I believe he garnered almost 30% of the vote as a third party. George Washington certainly could have stayed if he had wanted, but he was reluctant to serve his second term, and definitely declined to go for a third. An Americanist should have a more detailed answer for you.

There are a couple of close calls, but no one other than FDR has ever served more than 8 years. Probably the closest, and the only two people other than FDR to win the popular vote 3 times, were Andrew Jackson and Grover Cleveland.

During the election of 1824, different factions of the Democratic-Republican party, who had basically eliminated their rivals, the Federalists, put forward four different candidates. The end result was a bit of a mess and no candidate had a majority of electoral votes needed to win the election, despite Jackson winning the popular and electoral vote by a good margin. This was the only time the US Presidential Election was decided by the House of Representatives. When the election went before the House, only the top 3 candidates were considered, so Henry Clay's supporters threw their support behind John Quincy Adams to help him win the election. Understandably, people were not happy that a guy who lost by about 11% of the popular vote got the office. Jackson went on to win the following two elections outright.

The other case is Grover Cleveland who won the popular vote three times in a row in 1884, 1888, and 1892. In 1888, he won the popular vote but lost the electoral college to Benjamin Harrison because he lost New York and Indiana by a 1% margin. Reportedly, Grover's wife, Francis, upon leaving for the last time in 1889, told the White House staff to take care of the place because they would be back in 4 years. She turned out to be correct, as Grover won again in 1892.

As /u/suggestshistorybooks mentioned, Teddy Roosevelt did run for another term, however, it would have really only been his 2nd time elected, since the first was just serving out the remainder of McKinley's term. No one who took over for a previous president has ever won two terms after that, so 8 has been the most other than FDR. And as you mentioned, Grant made an attempt at a 3rd term in 1880, but the Republican convention was split between Grant and Blaine's supporters, known as the Stalwarts and Half-breeds (a contemporary term similar to when people call someone a "RINO" or "Republican-in-Name-Only" these days). They couldn't be reconciled but after 36 votes settled on picking Garfield as a compromise.