Assuming you mean only the Final Four and not the NCAA Tournament overall, the answer is UCLA.
Depending on which games you choose to include, the number of wins is a constant subject for debate. UCLA has 11 National Championships, the best record of any NCAA school.
One of the arguments over which wins to score stems from the 1980 Championship Game appearance against Louisville. Although the Bruins defeated Purdue (67-62) to reach the finals, the NCAA later vacated the win based upon alleged rules violations. Louisville beat the Bruins to take the National Championship, 59-54.
The other is whether one wishes to count the consolation games which were only played from 1946 to 1981. In those games the losers of each semi-final game played for the third place position. UCLA won twice (2 wins). In 1974 UCLA beat Kansas, and in 1976 UCLA beat Rutgers.
11 National Championships (22 wins in the final four, one win for the semi-final, one win for the National Championship);
2 semi-final wins without a win in the National Championship game (1980, 2006, 2 wins);
2 consolation wins (2 wins);
and the 1980 semi-final win which was later vacated (-1).
Now all you have to do is decide which wins you want to include and do the math.
Including the 2012 NCAA Tournament: Brigham Young University has made the Tournament 28 times and has never made it to the Final Four. Missouri is second, with 26 appearances without ever making the Final Four. Xavier, which so far has won in the 2012 tournament as this is being written, is third with 22 years without an appearance.
Penn in 1979.
From 1979 - 2010, 54 no 1 seeds have made the final four. That's 42.2%
1975 - The Indiana Hoosiers lost in the Mideast region's final to the Kentucky Wildcats.
This year's (2008) Men's NCAA college basketball Final Four is being held at: San Antonio, Texas.
The "Final Four" are the four teams that win the regional championships in the NCAA basketball tournament, thus becoming the final four teams in the tournament.
By making to the last four teams in the NCAA March Madness tournament
Georgia Tech has never won the NCAA Tournament. They were second in 2004 and made it to the Final Four in 1990.
The Final Four is the national semi-final of the NCAA Tournament featuring the winners of the four brackets, which decide who play in the National Championship
Brigham Young University has made 22 NCAA Tournament appearances without advancing to the final four. This is as of the end of the 2007 season.
louisville, oklahoma, stanford, uconn.
about 6 times
Georgia made it to the Final Four in 1983.
Including the 2012 NCAA Tournament: Brigham Young University has made the Tournament 28 times and has never made it to the Final Four. Missouri is second, with 26 appearances without ever making the Final Four. Xavier, which so far has won in the 2012 tournament as this is being written, is third with 22 years without an appearance.
2008
North Carolina, Connecticut, Villinova, and Michigan State
Penn in 1979.