Some fans who grew up in the 90s (as I did) will say that UF is our biggest rival. I would have said that in the 90s and 2000s, but I think it has gone back to being Alabama ever since UF began to decline after Urban left and Saban arrived at Alabama.
The UT/UGA rivalry only became a thing with divisional play in 1992 (as did UF), but the UT/UF game decided the SEC East and SEC Championship for a number of years in a way that UT/UGA did not.
I also think it is interesting that a significant minority of Alabama's fanbase will tell you that they consider Tennessee to be their biggest rival, not Auburn. It's nowhere near 50% (or even probably 20%), but a lot of Bammers will tell you they hate Tennessee more than Auburn, Fulmer in particular. Saban's comments on GameDay last year about him finding it hard to get his team up for Auburn in some years, but never for Tennessee, I thought were interesting too.