Bert and Ernie (from Sesame Street, not It’s A Wonderful Life)
They both seem to annoy the fuck out of each other and have wildly different personalities, but they stay buddies no matter what. Very sweet.
Hornblower and Bush from the Horatio Hornblower novels.
Sharpe and Harper from the Sharpe series.
Both are examples of fictional military superiors/subordinates who are also loyal friends who would die for each other (though Hornblower wouldn’t actually say as much. 
).
Rick Blaine and Sam… and maybe Rick and Insp. Renault, by the end of the movie.
Gilgamesh and Enkidu.
Dr. Bashir and Miles O’Brien.
Sheldon and Leonard, however much each annoys the other.
Lenny and George.
George: …well, we ain’t like that. We got a future. We got somebody to talk to that gives a damn about us. If them other guys gets in jail they can rot for all anybody cares.
Lennie: But not us, George, because I… see, I got you to look after me, but you got me to look after you.
The classic example from Greek mythology is Damon and Pythias.
Thelma and Louise
Cagney and Lacey
Laverne and Shirley
Starsky and Hutch
Shawn and Gus
It’s closer to a Frodo-and-Sam relationship than a partnership of equals, but:
Taran Wanderer and Gurgi from the Chronicles of Prydain
While Gurgi quickly latches onto Taran and repeatedly expresses his loyalty, it takes a while for Taran to recognize that he himself owes Gurgi the same unshakable commitment.
Dean and Cas?
Although, I haven’t watched the last half of this season so…maybe things changed?
Wallace and Gromit
See the TVTropes page on Heterosexual Life Partners; or (when the best-friends are of opposite sexes but still not romantically involved) Platonic Life Partners.
Darmok and Jalad at Tanagra.
Darby and Joan
Hal Jordan and Barry Allen
Daria Morgendorffer and Jane Lane
Jake Sisko and Nog
The entire main cast of One Piece.
Spongebob and Squidward.
Elvis:
Is that meant as a joke? Squidward HATES Spongebob. Occasionally, he’ll feel guilty about treating him badly, but he’s certainly not a “devoted friend.”
Johnny Storm and Peter Parker.
And McCoy (to both, and vice-versa)
Bugs & Daffy (I kid, I kid 
)
Chandler & Joey.