Chris & Dan
Ying and Yang like never before; a guy who pushes and a push-over. They're just really entertaining to watch, or hear I suppose since most of the humor is dialogue. It's great because while most of the humor is written, it still feels like a cartoon because of how exaggerated their personalities are. Also, their exchanges are hilarious ("WHAT DO YOU MEAN YOU WON'T KILL HIM! Why didn't you say this BEFORE?" "Well I was going to, but --" "But WHAT?" "-- but I didn't, you're right, I should have, I'm sorry..."). It's nice to have a duo that can swap between being decent friends towards one another, such as getting them out of tight spots, and then fighting right afterwards.
I think the last time we had a strong duo like that was Barney & Fred Flinstone. I would say maybe Mac and Bloo, but they didn't change in the episodes but rather in the seasons. In the early episodes, they were good friends. Later on, they became a cartoony Kyle & Cartman which got pretty boring.
Flapjack & K'nuckles
Flapjack's fascinations with K'nuckles is so adorable and funny and well written, it feels refreshing to see a duo with a genuinely caring character for his friend. Normally when a duo has a character who cares, the character ends up being dim-witted and making things worse. But Flapjack really does seem like he considers himself a close, close friend to K'nuckles, and in a funny, cute way not in an obsessive, fanboy way.
I also like how the writers treated K'nuckles like he was weirded out, but still acknowledged that he needed Flapjack, as he's his only friend. They didn't write him like Squidward and they didn't keep repeating the same thought process where K'nuckles only realizes he needs Flapjack after being alone, once or twice yeah, but later on, they made it so he realized ahead of time that Flapjack is his only real pal, so that he needs him and shows a bit of care in losing him.
Doc Brown & Marty McFly
I realize they're not cartoons, but after playing the new TellTale episode 3 today, I just had to mention them. I don't know what it is, but seeing these two on screen (on monitor?) together just makes me happy. Maybe it's that the individual characters are really good, so putting them together just makes it better, I dunno. They're not opposites, they don't have hysterical dialogue exchange, but they're believable friends with distinct personalities/mannerisms/etc., so it's just really nice to see a considerably down to Earth (Compared to the characters above) duo in crazy situations.