So:
33 Harry Truman (D-Missouri) 1945-1953
Truman was the last President who had the possibility to run for a third term, as 22d Amendament didn't apply to the incumbent President, but he declined as he saw zero possibilities to win.
34 Dwight David Einsenhower(R-New York) 1953-1961
Okay, Ike was widely popular and he didn't like very much JFK and Nixon but he was 70 in 1960, so very old for his time, and it seems very clear that Mamie didn't want be a widow of a President.
35 John Fitzgerald Kennedy (D-Massachusetts) 1961-1963
We all know why, unfortunately JFK was not able to run for a second term, so it's pretty impossible aspire to a third one.
36 Lyndon Baines Johnson (D-Texas) 1963-1969
Lyndon was a pretty popular President before Vietnam but after he was so unpopular that Secret Service told him it was not able to protect him if he had wanted run for a second term. So nay.
37 Richard Milhous Nixon (R-California) 1969-1974
Tricky Dicky surely wanted to reign at life but Watergate was fatal for him so, sorry Dick, it will (not) be for an other time.
38 Gerald Rudolph Ford (R-Michigan) 1974-1977
He lost reelection and didn't start any comeback so his door toward White House was closed in 1976.
39 James Earl Carter (D-Georgia) 1977-1981
Same thing for Ford.
40 Ronald Wilson Reagan (R-California) 1981-1990
Yep, Reagan had some signs of Alzheimer but he was not ill and declined reelection for one of the most popular American President would only confirme that a mental problematic person governed the US for eight years. So he run but when Alzheimer becomes clear during a international crisis (Panama or Kuwait, I suppose) he is removed due 25d Amendament and replaced by Bush Sr.
41 George Herbert Walker Bush (R-Texas) 1990-1993
Maybe he didn't say "Read my lips" but it could be worst as he will be seen as a traitor of Reagan policies and a tricky person who took advantage of the illness of an old sick man. He lost reelection.
42 William Jefferson "Bill" Clinton (D-Arkansas) 1993-2005
He is young, he is popular, his scandals didn't count as "It's economy!". Gore could have won if he didn't distance from Clinton and had taken a more relaxed and less arrogant style, so Bill could win in 2000. His response at 9/11 Attacks would be controversial as Republicans could point as he didn't kill Osama in Sudan when he had the occasion to do it. His heart attack in 2004 will be also a big problem and probably GOP will be able to take the White House.
43 John Ellis Bush (R-Florida) 2005-2009
The GOP Presidency will escalate foreign wars but the Recession will destroy its reputation.
44 Barack Hussein Obama (D-Illinois) 2009-...
Obama was-well, actually is- widely popular and could easily run and win a third term thanks to popular image, high black turnout, connections with young vote and a more middle-man appeal than Hillary. He is the current President and many are thinking if Barack will try to match FDR and run for a fourth term or if he will retire, probably in favor of his Secretary of State Hillary Clinton.