Mississippi State has historically been Alabama's whipping boy.
Observation:
We have beaten Mississippi St. 72 times.
Texas has beaten Baylor 73 times, however they lost to them this year.
So far as I can tell, if we beat them this year, we will have beaten them as many times as ANY team has ever beaten another team in Div 1A college football.
I couldn't find any team with more than 73 wins against an opponent***. My search was not exhaustive. I only checked what would appear to be obvious match-ups.
I found some other perennial rivalries that were close:
Notre Dame has beaten Navy 70 times (they lost this year)
Michigan has defeated Minnesota 70 times (they don't play this year)
***EDIT: I just found where Oklahoma has beaten Oklahoma St. 80 times
*** EDIT2: OK. I was way off. Nebraska has beatean Iowa St. 85 times, Kansas 90 times, and Kansas St. 77 times.