LSU and Alabama should become one of the best thanks to Saban and the high level that each of the teams play on.
It's definitely up there, but it's been one of our chief rivals since even before than. The problem with that is LSU will probably never have just one true rival, just a lot of really strong secondary rivalries.
We play Arkansas every year after Thanksgiving, but nobody really gives a crap about that series, which only started 16 years ago. In the 50s and 60s we had an intense rivalry with Ole Miss, but they are no longer the powerhouse they were. While they often make it interesting they rarely beat us, so I can't consider it that much of a rivalry. Though there is a healthy amount of dislike between the two schools. Then there's the Egg Bowl they have with Miss St. which occupies most of their hatred.
Then there's Florida, who we play every year and are very heated with.....Only problem is we're like 4th or 5th on their list of rivals behind FSU, Georgia, and Tennessee.
Auburn is also one of our main rivals and you could say that on the field they are our most competitive rival. This matchup always has huge division title implications and almost always comes down to the final second. Unfortunately, they have Alabama as their main rival.
Basically, every school's fans hate LSU fans, and LSU fans hate fans of every other schools, but don't hate one particularly more than any other. Therefore, LSU has several hated foes but no true rivals.