The more democratic party political era that began with Andrew Jackson in 1828 established a new pattern of most chief executives serving single terms (or, if they succeeded to a death-created presidency, no subsequent election). It took special qualities and special circumstances to win reelection: Jackson himself; Abraham Lincoln and Ulysses Grant in the Civil War-Reconstruction era; Grover Cleveland as a Gilded Age reformer (though his were electoral victories separated by a Republican presidential term); the anodyne William McKinley; the Progressive Democrat Woodrow Wilson.
From Jackson (1828-1832) until the coming of FDR in 1932, only six presidents were elected to two terms—and 18 elected to one (or, in the cases of John Tyler, Millard Fillmore, Andrew Johnson, and Chester Arthur none).*
To what degree did FDR's four elections change the pattern? Since then, two-term presidencies have been more frequent: Dwight Eisenhower, Richard Nixon, Ronald Reagan, Bill Clinton, George W. Bush, Barack Obama. As before, those who came into office as vice president when the incumbent died (or, in Nixon's case, resigned)—Harry Truman and Lyndon B. Johnson—did not pursue the reelection for which they were legally qualified.* And as before, a less than stellar record, or a vote-scattering third party candidacy—Jimmy Carter, the elder Bush—meant that the incumbent fell by the wayside.
And the most recent two-termers—George W. Bush and Obama—won less than smashing triumphs. Obama, indeed, was the first two-term president in modern times to win his second term by a considerably smaller margin than his first one.
So I think it may be said that in its long history the presidency has generally been hostile territory for extended presidencies—with the exception of FDR. But his four elections went so against the American grain that the 22nd Amendment embedded Washington's two-term limit in the Constitution.
What may we look for in the Age of Trump? Given his low poll standings, his increasingly shaky position in his own party, and his inability so far to expand his appeal beyond his core, Trump's second-term prospects (both for nomination and reelection) are less than bright. Of course, we're only six months into his first term, but it is difficult to see much light at the end of the Trumpian tunnel.
Given the past record and this future prospect, the need for fresh faces and a fresh approach to governance in 2020 looms ever larger. Trump may have activated a portion of the electorate that Hillary Clinton (who in her way was almost as deplorable a candidate) missed. And yet, Democrats are so far not making very effective noises about reaching out to these voters, much as the Republicans fell short of its efforts to engage Hispanic voters after 2008.
Who might those new faces, and what might those new approaches, be? The Democrats might be well advised to skip over their current leaders—Clinton, Bernie Sanders, Joe Biden—and not-so-young aspirants like Elizabeth Warren. As of now, none of these set many pulses racing. It might be best for the Democrats to see who emerges in the wake of the 2018 election.