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Traditionally, Georgia. 
They had the better program, but UGA is why Florida famously had no SEC titles pre-Spurrier.
Every single time Florida got late into the season in 1st or near 1st in the conference, Georgia would beat us.
From the 60s into the 80s.
.
More along the lines of what the question was intended, Auburn.
Multiple Auburn teams that were ranked lower or not ranked beat a #1 or top-5 Florida team.
93, 94, 01, 06, and 07
Those years, Florida was ranked 4, 1, 1, 2, and 4 and lost each time.
The 06 game stands out, as AU won 27-17 without an offensive TD.
FG, safety, FG, FG, blocked punt TD, FG, fumble return for TD (last play)
That was stupid.

They might qualify after all.  While they were good historically, we were close to the same level. 

All-time win%
UT:    .673  (11th)
LSU:  .655  (14th)

Conf. Championships
UT:    16
LSU:  16

All-time wins
UT:    886 (10th)
LSU:  852 (12th)

Based on that, you probably wouldn't expect a 10-21-3 record against them, but that's what it is.  And that includes an outlier period of 5 straight wins beginning in 2006.  So think about this, in the first ~80 years of playing each other, LSU accrued just a 5-20-3 record against them.  Not nearly as close as their records might suggest. 

Tenn had a couple Dark Ages as of late. One thanks to Bill Battle and the length of time it took Johnny to start getting some players/ Then the fiasco's that followed Phil.

LSU at night is a difficult win. I think that you are one of the teams that Tenn. normally gets up for and your team may overlook them some

They are usually good games though, no matter the sport.