The 1871 college football season is the only year since the first season in 1869 in which no games were played.
This is likely due to the difficulty in scheduling games for the then-new sport. Only two games had been played in each of the previous seasons (1869 and 1870) by only three teams (Rutgers, Princeton, and Columbia).
Princeton did play several games against a Princeton Theological Seminary team in 1871, although the Tigers considered these "practice" (or exhibition) games, and the general convention is not to count these as official games.
Because no games were played, 1871 is the only year since college football play began in which no college football national champion has ever been named, retroactively or otherwise.