Here are the presidents impacted by the 22nd Amendment's two-term limits clause
Dwight Eisenhower
He would have likely deferred back to tradition and not have sought a third term. If he did, he might have done better than Nixon but certainly Kennedy was running on a change platform. And how would the televised debate looked if it was JFK vs. Ike instead of JFK vs. Nixon?
Ronald Reagan
He was still popular which is why HW Bush was able to win in 1988 but at the same time Iran-Contra might have been more a focal point in the campaign. If the Ds still nominate Dukakis, expect Willie Horton to appear. Lee Atwater was a player in Republican politics at this time and likely would have been involved in a third Reagan general election bid.
The wild card is this: Reagan's health. He left office when he was almost 77. Four more years in the White House likely would have caused his decline quicker. In the real timeline, he disclosed his Alzheimer's in 1994 and retreated from public view until his death ten years later. The campaign would have kept anything about Reagan's health a secret. If it became serious enough, it might have triggered the medical incapacitation clauses of the 25th Amendment (or whatever amendment it is called since there is no 22nd in this exercise) resulting in HW Bush (or whoever his VP would be) as POTUS for a period of time.
Reagan might have taken a page from FDR who in 1940 was mum about seeking a third term and instead chose to allow the Democrats at the convention to have their say. FDR nearly won the nomination by acclimation but in the running mate balloting, the delegates picked Henry Wallace, FDR's Agriculture Secretary. Reagan likely would have seen Kansas Senator Bob Dole, Indiana Senator Dan Quayle, and other Republicans challenge him for the nomination. Or perhaps his sitting VP challenge him in a rematch of 1980. It would not be out of the realm if Reagan dumped Bush for another VP since there was some friction between the two that stemmed from the 1980 primary.
Bill Clinton
Likely yes to seek and probably would have won a third term in 2000. I do not see Clinton making the same mistakes that Gore made towards the end on the campaign trail as well as taking the R nominee (likely Texas Governor George W. Bush) very seriously and surgically taking his arguments apart.
Would have MonicaGate come up? Certainly, but as shown in the 1998 midterms, the Ds survived because the public saw what it was all about given that Clinton's job approvals remained high throughout the ordeal. I think the public did not want to re-litigate that again.
Clinton wins in 2000.... boy that starts a chain reaction to a lot of what-ifs...
Is there a 9/11? Which likely means no Afghanistan, Iraq, Gitmo, PATRIOT Act
There are no Bush tax cuts, no financial deregulation so no 2008 financial crisis
What role does Hillary Clinton play in a Bill Clinton third (or even fourth term)?
Someone mentioned the balanced budget, does that allow for Bill to be more bold in a third term such as go for health care reform?
Bill's health: he did have heart surgery in the early 2000s. Again the presidency does put a strain on a person.
Is Barack Obama still playing in Illinois politics or has he retreated back to academia?
What becomes of the Republican Party?
One constant is that Donald Trump is in the public view in some capacity...
George W. Bush
No and if W Bush was on the ballot in 2008, he probably would have done worse than McCain. Remember his approvals as he was leaving office was in the 30s. Imagine the financial crisis happening as Bush is campaigning for a third term.
Barack Obama
Obama would have gone for a third term despite charges from the opposition that he was becoming a dictator (which was what FDR's opponents said in 1940) and likely would have won. Obama left office with approvals in the high 50s and during the 2016 campaign polled more favorably than Hillary Clinton and Donald Trump.
Maybe would have dropped Biden in favorable for someone younger and could carry a key battleground state or even expand the map. And again, as someone mentions, it depends on who Obama drew from the Rs. In 2012, the only R that the Obama camp feared was John Huntsman, the former Utah governor & US Ambassador to China. Certainly the Obama team would have taken every R seriously, including Trump.
Financial deregulation didn't pass under Bush. Most of that happened in the 80s and 90s. Bush actually wanted to regulate Fannie Mae and Freddie Mac.
Anyway, we should not forget that (almost?) nobody sought a third term before Roosevelt. Coolidge for example was extremely popular when he left office and he easily would have won reelection, but he didn't seek a third term.
Clinton, Obama and Reagan were the only ones that might have ran for a third term imo. But Reagan's age probably would have convinced him not to run. Obama seems like a guy who would respect the tradition and in April 2015 (when Obama would have had to decide) Hillary was looking like a decent candidate. But there is a possibility that he would have ran, and he would have won a third term in that case. Bill Clinton probably would have sought a third term, and I think he would have won.
