Since your question is "who is LSU's biggest rival?" not "with what team does LSU have the biggest rivalry?" or, "for what school is LSU the biggest rival?" I will take a stab at it from the standpoint of LSU's rivals. But that depends of course, upon how one defines "rival." Is it teams against which LSU has played the most games? How about the team against which LSU has the most even record? I submit that it is a combination of the two. But, how many games to qualify? If we were to say LSU must have played the team a minimum of 75 times, then there aren' t too many from which to choose, but it would be Ole Miss. We have won 59% of our games against them, overall. But, if the thresshold was set at 40 games, it would be Auburn (followed by Florida and then Texas A&M - but we haven't played them in a while). These are the teams LSU has met on the gridiron most frequently: Opponent W L T W % Mississippi St 63 33 3 66% Tulane 65 22 7 75% Ole Miss 53 37 4 59% Alabama 21 43 5 33% Kentucky 37 15 1 71% Florida 22 27 3 45% Arkansas 32 17 2 65% Texas A&M 26 20 3 57% Auburn 21 19 1 53% Tennessee 5 21 3 19% Vanderbilt 19 7 1 73% Georgia 14 11 1 56% Louisiana-Laf 21 0 0 100% Louisiana Tech 16 1 0 94% We may not be Auburn's biggest rival, but, to me, Auburn now has become our biggest rival. There are other, similar situations. For example, Texas is Texas A&M's biggest rival, but OU is Texas' biggest rival.
After Landry and the Auburn defense got done with Alabama, the headline was "Bama is on their Back". Cling to that 10-2 record and Number 8 ranking. It'll be a memory soon enough.
Used to be Ole Miss, and still is to some... But now I'd have to give Auburn a strong vote for that one in lieu of the last several years games. Behind Auburn, but gaining ground is the referees.
It would certainly work out perfectly if Arkansas became LSU's fiercest rival. The Hogs don't have a single major rival since they left the SWC, and we don't have one either. For that to happen, however, both teams have to be in contention every year, or else something has to happen to enrage the players and fans of both teams so that they hate each other. So far, that hasn't happened.