What would have happened if George Washington had run for a third term? No doubt he would have won. Would he have still died two years later causing John Adams to take to office as president? Would presidents be legally allowed to run for a third term nowadays? What butterflies would ensue? What would happen?
Since his OTL death in 1799 derived from exposure to the elements followed by pneumonia, it is sure that a third term would have butterflied it away. Most likely, Washington would finish his third term.
One of the most likely butterflies (one I used in my Ameriwank TL) is that he would have very likely stopped the Alien and Sediction Acts, which pretty much crippled the Federalist Party for good. He stood for neutrality, but he also stood for defending America effectively, so he he would have still fought the Quasi-War with France (maybe with a formal declaration of war to France). Adams would have almost surely won the 1800 election, quite possibly the 1804 one, too, and Federalist victories in 1808 and 1812 with Pinckney or Hamilton are a definite possiblity. Thus, George's third term could easily butterfly away the disastrous Jefferson Presidency. If this happens, and the Federalist Party remains in power, they can use the Quasi-War and the First Barbary Wars as justifications to build a strong US Army and Navy. Certainly they would do so as a response to growing British harassment of American trade, instead of Jefferson's moronic self-harming embargo.
With a strong US military in the War of 1812, substantial territorial gains in Canada become plausible. Maybe Lower Canada/Quebec itself or maybe not. But Upper Canada/Ontario is definitely in the cards. If America gains Upper Canada (and quite possibly Lower Canada, too) in 1815, it is all but sure that in the long term, Rupert's Land, Northwest territory, and Pacific Northwest are fated to become American, too, conquered or bought by the USA during the 19th century, as a combination of factors (poor strategic communication with Britain and rump atlantic Canada, if it exists at all, scarce British and abundant American settlers, and declining fur trade) makes them look more and more not worth the economic and military effort and risk of another major war with the USA to keep them British for London.
Another big butterfly is that keeping the Federalists longer in power would allow Hamilton's economic program (infrastructural improvements, subsidies to banking and manufacturing) be implemented for another decade (a longer Federalist dominance surely butterflies away the deul for Burr and keeps Hamilton alive, too) makes America's economic development be even quicker and stronger than OTL. It is kinda more difficult to say how their longer dominance would have affected the budding sectional antagonism between North and South.
There is a definite possiblity that the first US party system would endure up to modern times, as the Federalist Party keeps the place that historically was filled by the Whig and Republican Parties in turn. Surely, incumbent fatigue would eventually bring the Democratics in power, no matter how successful the Federalists may have been, but if they win the War of 1812, they gain a substantial extra lease on power and their economic policies were sound, so the Jeffersonian democratics never gain power, the moderate new breed led by Madison and Monroe may or may not ever do, quite likely the change in power happens with the Jacksonian revolution and the advent of mass parties in 1820-1828.
Washington's third term would have surely established an unchallenged precedent. For the rest of American history we may look to many successful two-term Presidents that are not crippled by ill-health or scandals seeking a third term and most of them getting it, too. However, the likely gain of most or all of Canada in 1815, the longer Federalist dominance, and the different party system creates far bigger butterflies in the list of presidents than a three-terms standard may ever do. One thing is certain, however, if the 22nd Amendment even happens ITTL, it shall mandate a three-terms limit.