Through 2012, North Carolina and the University of California-Los Angeles (UCLA) have each played in 18 Final Fours, but due to the use of ineligible players, UCLA's 1980 appearance was vacated by the NCAA governing body. Thus, the record books reflect that North Carolina has made 18 appearances and UCLA 17.
To answer your question very precisely, however, the two teams that "have played the most" in the Final Four are UCLA, which has played 33 games in the Final Four (18 semi-finals, 13 championship games and 2 third-place contests) and North Carolina, which has played 30 Final Four games (18 semi-finals, 9 championship games and 3 third-place contests). Even when one subtracts the vacated appearances, UCLA has played 31 Final Four match-ups to North Carolina's 30.
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