When the new American nation was deciding on the powers and limits of its executive, there was a heated debate. Many wanted a powerful, almost regal, presidency that would give the new government credibility in the world. Others wanted executive powers to be restrained and limited by Congress. A sort of compromise ensued that was put to the test by the first occupant of the office, George Washington. Famously, after serving two terms, Washington chose the “Cincinnatus” approach and refused to run for a third term. Wrote Washington to a friend, "Prudence on my part must arrest any attempt of the well meant, but mistaken views of my friends, to introduce me again into the Chair of Government." He walked away and set an informal precedent for his successors.
After many later “imperial”, Caesar-like presidencies (e.g., Abraham Lincoln during the Civil War and Franklin Delano Roosevelt, who was elected to four terms), the Constitution was amended in 1951—prohibiting any person from being elected to the office more than twice. So—at least in terms of succession—the Cincinnatus model has prevailed over that of Caesar.