Trivia answer.
Thanks for comments. Yes, this is just about football. Yes, it was meant to include any game ever between the teams, even prior to conference membership for any team. No, I didn't consider ND as a conference team. Maybe I should. They're kind of like a distant cousin. Are they family, or are they just a party crasher?!! Going forward, I'll make clear whether they are in or out, as least as far as the trivia questions go.
I had a hard time coming up with the wording of this question, and apologize that it wasn't so clear. In any case, here is the answer.
To the poster who responded 13, I feel bad that I left that trap for you. 13 is the only answer that is not possible. If each of 13 teams played each of the other 13 teams, then perforce the 14th team would have had to play all 13 of the other teams. Another way to look at it is, if one team, say team A, played every other team except team B, then BOTH team A and team B would not have played every other team. Enough of the professorial logic.
Louisville has played all but Georgia Tech. We've played just once against Duke, and, of course, Clemson for the 1st time last season.
There is one other pairing that has not yet occurred, and that is Pitt vs Wake Forest.
So the short answer, after all that prose, is:
10