It's amusing to watch the "which rivalry is best?' contests staged every year by ESPN and the usual suspects - in reality they are nothing more than advertising blurbs since the contest is "sponsored" by some megabrewer pushing their latest horrible-tasting mass-produced concoction.
You might call it a rivalry between rivalries except it's not. The best/strongest/most intense rivalry is Michigan-Ohio State (Ohio State-Michigan if you prefer) and as sports talk radio is wont to say "it's not even close."
What is required for a strong rivalry?
1) A large, insanely loyal fan base. When you have two huge Big Ten institutions the students and alumni alone provide an ample population. Throw in the majority of residents in each respective Rust Belt state and you have two opposing armies with "foreign legions" in every state and every corner of the world.
2) Two unique identities. The schools, the states and the people provide enough of a contrast but the Woody-Bo years intensified the rivalry and made it forever strong.
3) Success outside the rivalry. Other rivalries have teams with plenty of underwhelming seasons, including Alabama (say the words Shula or Dubose and watch them twitch violently), Auburn, Stanford, Cal, Pitt, West Virginia, and of course Army and Navy. Ohio State and Michigan might have 10 lean years combined since 1960. Appalachian State and Illinois notwithstanding OSU and UM find themselves in contention for the Big Ten title yet again.
4) Transcending the sport. Let's push Auburn-Alabama to the top of the list for a moment. The biggest drag on this rivalry is its intrastate nature. Oh sure there's more than enough banter, bitterness and outright hate to fuel a year-round feud but the fan bases must interact on a daily basis. And...the football game is always the most important thing in AU-UA. Michigan-Ohio State transcends the sport because it engenders/perpetuates dislike between two entire states. Apocryphal stories about Woody Hayes pushing his car south across the border so he wouldn't have to buy gas in Michigan aside, a Michigander meeting an Ohioan (and vice versa) will elicit a squint and a size-em-up reflex every time. When either visits other states, even foreign countries they are asked about the rivalry. Doesn't happen with any of the rest (although Cal/Stanford fans may still be asked about the Lateral Game).
Thus Ohio State-Michigan wins almost by default. It has too many advantages and almost no disadvantages. In successful years (i.e., national contention) Michigan's interest has occasionally wavered as they faced a Notre Dame or another tough out-of-conference opponent but during the leaner years they certainly look forward to the OSU game as a chance for redemption or simply spoiling another's parade. Ohio State could play the New England Patriots for the first 11 games of the season but their focus would still be on Michigan (just ask John Cooper). That is obsession. That is monomania. That is passion. That is a rivalry.