Final answer:
George Washington didn't run for a third term because he chose not to, setting a voluntary precedent that other presidents followed until the Twenty-Second Amendment made it a constitutional requirement.
Step-by-step explanation:
The reason why George Washington didn't run for a third term after 1792 was simply because he chose not to run. Washington set a precedent that was followed by subsequent presidents, who limited themselves to two terms in office. The decision not to seek a third term was Washington's personal choice and was primarily influenced by his desire to retire to private life, the growing political partisanship, and his wish to set an example for the regular rotation of public officials. It was not until after Franklin D. Roosevelt served more than two terms that the Twenty-Second Amendment was passed in 1951, which constitutionally limited the presidency to two terms.