From
2001-
2002, the answer was 71.
2006, 25's robot made it through New Jersey, Las Vegas, and almost Saturday morning at Championship without losing once.
2010, 469 had a dominant strategy, and 1114 made it through three regionals undefeated and won all but one qualifying match at Championship. (With the seeding system that year, that "loss" was really how they locked up #1 seed...so you tell me.) They then marched through the Championship eliminations together, not losing a match before being swept in the Einstein finals.