Origins: Alistair and Morrigan's contention, followed closely by Morrigan and Leliana, and Dog and Sten.
It's always hilarious watching Alistair and Morrigan fight, and it's definitely a highlight of the game. I also love how Leliana started her relationship with Morrigan by attempting to go clothes shopping. Of course, Sten and Dog's weird respect for one another is really funny. Honestly, pretty much any combination of companions is awesome.
Awakening: Oghren and Sigrun.
Sigrun has some hilarious lines towards Oghren. I kinda think that Oghren's dialogue improved in Awakening, which probably helped his conversations with Sigrun. Though, again, any combination of companions is hilarious.
2: Aveline and Isabela, followed closely by Merrill and Isabela, and Anders and Fenris.
When Aveline and Isabela get together, hilarity is always on the horizon. I think it's funny listening to Anders and Fenris' debates, except it makes me feel bad whenever they try to bring Merrill into it. One of the things I love about Merrill is her contrast to Isabela. She wants to be like Izzy so much, even though she doesn't have it in her. They are almost like sisters, really. Varric and Isabela make any banter with any companion hilarious, so there's always going to be something to look forward to when you bring 1 or 2 of them along.
Inquisition: Varric and Cassandra.
The way they fight, they're almost like siblings or a married couple. I love it when Varric pushes Cassandra's buttons (in a non-sexual way). I also love Dorian and Sera's banter together. Dorian usually inadvertently annoys Sera, but always takes her retorts well. Once again, every companion combination is interesting.
Overall: It's hard to go past Alistair and Morrigan, but if I had to choose someone else, probably Varric and Cassandra.
Side note: I reckon that Merrill and Isabela are probably the two closest companions in Dragon Age. Merrill brings out Isabela's soft side more than any other person can (excluding Hawke)... even though all she does is be naive (and cute). Most don't get along too well, and the ones that do are often one-sided (e.g. Dorian likes Sera, but not the other way around), or it's more of a sexual relationship than a friendly one.