I think the rivalry between the South Central Upper Ohio College of Medical Accountants and the Portsmouth University of Jamaican Studies, Zanesville Campus is the biggest in all of sports. Sure, the SCUOCMA White Coats haven't beaten the PUJSZ Reefers in a dozen years, but they've played some really close games. Look at the scores of the last ten years: 10-7, 19-7, 44-7, 44-39, 39-37, 100-4, 7-4, 4-3, 10-3, and 10-8. 7 of the 10 games were pretty close. And, of course, it made the local newspapers (page 4) when "Stethy," the Stethoscope, the mascot for SCUOCMA sucker-punched the Reefer mascot, Dave. The next year, the Reefers almost got their entire team disqualified for unsportsman-like conduct for stomping on the White Coat logo at the 50-yard line. These two teams absolutely hate each other, and have for at least 80 years.
The two teams started playing each other in 1924. The PUJSZ Reefers, who were still located in Portsmouth (until 1933), won 3-0. The White Coats bitched about the Reefers running up the score. The Reefers won the next 6 meetings by an average score of 17-2. The White Coats finally won in 1938, 19-6, with the White Coats' hero, Reggie "The Otter" Triffit scoring three touchdowns. The White Coats then won throughout the '40's, but the Reefers refuse to admit any of those losses, citing that their team was destroyed by the military draft.
Throughout the '50's, they traded wins, with the visiting team winning every game. Tensions rose another notch in 1957 when the Reefers won with a last-second field goal 33-31, and the Reefers' fans sat back and relaxed. The White Coats' fans were disgusted, thinking the Reefers should tear down the goal posts, and act like assholes.
The worst the rivalry got was in 1992, the last time the White Coats beat the Reefers. The Reefers were so pissed about the 29-3 game that the players went into the White Coats' locker room and gave them all titty twisters.
That's my vote for best rivalry in all of sports.