Packers vs Bears - people forget how close the rivalry is for over 100 years because of how dominant the Packers have been the last 30 years. Packers 106 - Bears 95, and 6 ties.

Cubs - Cardinals. While the Cardinals have more championships, it's the Cubs that actually have more head to head wins dating back to the 1890s.

I can't really pick a 3rd because I have no rooting interest, but college football rivalries like Florida/Georgia, Texas/Oklahoma (playing right now), or in basketball Duke/UNC are always interesting to watch on a lazy day.