When it comes to football there is TTUN and everyone else. Sure, the rivalry status of "not our rival" Penn State gets brought up every now and then, but it just doesn't hold a candle to our hatred for TTUN.
Other schools certainly have multiple football rivals though: USC has ND and UCLA. FSU has Florida and Miami. UGA has Florida and GT. Even TTUN often claims three with OSU, MSU, and ND.
Yes, we hate Michigan in all sports, but the gap between Michigan basketball and other teams that give us a sour taste is no where near as large as the gap between hatred of Michigan football compared to all others.
1. Michigan - though both OSU and Michigan are typically considered top 15 basketball programs all-time, our eras of success haven't always overlapped - 1964, 1992, 2012, and last year are the big time exceptions that come to mind. That said, the Fab Five alone would be enough to put the Wolverines atop this list and there's something about those gaudy maize jerseys that makes my skin crawl. The truth is most things about Michigan make my skin crawl so no matter the sport, I relish any W over them.
2. Indiana - OSU has played no school more since 1950 and there was frequently a lot on the line when the two met. Only Purdue has more Big Ten titles than the Bucks and Hoosiers. While IU has gone 5-1 in NC Games, OSU has posted a 1-4 record...otherwise both have had (surprisingly to some) similar success with each having 22 Big Ten titles (each claiming 13 outright), both finishing the regular season number 1 in the AP poll an equal 4x, and both representing the Big Ten often in FFs, OSU having 11 and IU 8. Throw in Bobby Knight's history and some amazing games over the decades and this one ranks high for me.
3. MSU - Undoubtedly some recency bias here, but during the Matta era, Izzo and co. were right there most years atop the Big Ten. Since the BTT began in 1998, the Spartans have won it 6x, the Buckeyes have won it 5x, and no one else has more than 3. MSU has 10 regular season titles during that same time, the Bucks have 7, followed by Illinois and Wisconsin each with 5. Our 2010 and 2012 Big Ten titles required great wins at the Breslin Center and we've met the Spartans in the BTT 8x.
4. Cincinnati - basically the opposite of recency bias. Ancient bias. But it's hard not to hate a program that spoiled a 27-0 perfect season in OT of the NC game that otherwise would leave the 1961 Buckeyes in the conversation as the greatest team of all-time. OSU went 53-3 in 1961 + 1962 with zero NCs because 2/3 of those losses came to UC in the NC Game. The rivalry has been one of avoidance as a result of disdain until the two met in the 2007 Wooden Classic, the 2012 Sweet 16, and finally just recently in a home-and-home with OSU taking all 4 of those. We've looked at UC with little brother status and left it at that over the years, but this would make for one hell of an annual matchup or annual season opener.
5. Kentucky - I'm listing this as a personal response to the question of "outside of the Big Ten, which team are you most excited to see the Buckeyes beat?" So maybe this would be like the Indians/Guardians or Tigers claiming the Yankees as a rival, as obviously UK is cemented as a basketball blue blood and likely in no way considers OSU any type of significant rival. But OSU has faced the Wildcats 6x in the NCAA tournament and holds a 5-1 record against Big Blue. Unfortunately, if not for that painful loss in the 2011 Sweet 16, that record would be 6-0 and we may have a NC from 2011. History is flipped outside of March with UK holding a 5-2 record, which was 5-0 until we began meeting in the CBS Classic. A reach? Sure. But outside of the Fab Five and the Bearcats in '61 and '62, I'd place that 2011 matchup as our most painful NCAA tournament loss and we've certainly been a March thorn in Big Blue's side otherwise with the series basically dead even.
Honorable mention: Bo Ryan and his buzzcut Badgers.
Closing side note: it is an absolute crime that the Big Ten doesn't grant each time a couple of locked basketball rivals for home-and-homes. Michigan and IU should be on our home schedule annually...ideally with MSU as well.