This is a non-specific question. There are four genuine national champions every year determined through playoff systems. Divisions I-AA, II, III, and NAIA. There is one national champion determined each year through a combination of voting and computer rankings: BCS.
However, if you don't worry about that distinction between championships which are determined by votes and championships which are determined by actual competition on the field, then the answer is Princeton University with 24 championships.
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