What it says on the tin. What if George Washington had run for and won a third term? Assuming of course that his death in 1799 is butterflied away (not hard, the circumstances surrounding his death were rather weird) and that he lives long enough to finish it.
Note that he probably wouldn't have been running unopposed this time around. Opposition to Washington's Federalist policies (I know he was an Independent, but he governed like a Federalist), had solidified and it was likely that they would have run a candidate (presumably Jefferson) against Washington. In fact, some believe that this played a part in Washington's decision to retire. He didn't want to tarnish his reputation by fighting to retain power. Anyway, even though it's not unanimous, he would still win in a landslide over Jefferson.
How would the third term go? The main issue during the OTL Adams administration was the deteriorating relationship with France (XYZ affair, Quasi War, etc.) and the administration's domestic reactions to that (Alien and Sedition Acts). How would Washington have handled these things differently? Also, how would our modern perception of Washington be different with him running in an actual election with an opponent and then dealing with significant opposition during his third term?
If he retires after a third term, who would win the 1800 election? My guess would be Adams. He was the Federalist favorite and Jefferson will probably have taken a hit in popularity after running against Washington. The Republicans (or Democratic-Republicans, as modern historiography insists on calling them) will probably nominate him anyway, but he'll lose. If Adams is President, would he still make the Louisiana Purchase like Jefferson did? I'm not sure what his position on it was. I know that in OTL, the Purchase faced opposition from both Republicans (who felt it was unconstitutional to buy land like that) and Federalists (who feared that the acquisition of so much rural territory would hurt their political prospects). I'm not sure where Adams came down on the issue. In 1804 the Republicans will probably nominate either Madison, Burr, or Clinton.
In the long term, how would America's political culture be affected by having a three term tradition rather than a two term one? Obviously most of the OTL American political figures have been thoroughly butterflied, but I'm more wondering about the general impact of popular Presidents staying in office for 12 years rather than 8.