I've gone through the game I think eight times now? The Hillbilly has always been in my team. He is the least useful outside of his own area (compare this to, say, the Monk, who can solo a number of puzzles), but I always enjoy romps through the carnival.
The TF2 items may have biased me towards making him my favorite, but he's always welcome in my expeditions.
The Knight area is kind of short and easy (even easier if you use the Monk), but still very funny and the Knight himself remains useful throughout if you are impatient with ladders or just want to not worry about interface flubs when trying to drop lit dynamite.
The monk may or may not be a game breaker in some places, but his area's puzzles tend to largely not involve the others and that's kind of dull, even though his brainteasers are very good.
Those were the three I used to do the no death run after a few tries.
The adventurer's story kind of explains itself too soon I think but her platforming-based stuff was a welcome breath of fresh air to me. I didn't really notice any places where she provided any serious puzzle shortcut benefits outside of maybe the island.
I really like the scientist's area, and she herself is decent for taking one or two character switches out of the equation in some places. I also like that she somehow weighs less than the adventurer (but she is shorter, so I guess that contributes to it). The puzzle involving figuring out the current day and which code was correct was probably a highlight, and yes, the wet floor sign was amazing.
The twins were cool but it was mostly a solo act until the very end. I guess that can't be helped, since they had to use the ghost ability in their area somehow. I don't recall many places where their ghosting really helped much other than maybe preventing the need to use a third character in one part of the zoo.
The time traveler I liked in terms of complexity, even with the bits of backtracking. The bucket thing wasn't so much an issue, it just took me a while to remember I could have a character hold onto something and then switch, haha. I could really only think of maybe one area outside of her zone where her ability provided a shortcut the monk's couldn't, but this may be my ignorance.
Really I can't think of any part I didn't like much other than the island and the monk's area. Maybe it was the parrot's voice that kind of grated on me a bit, the voice sounds a bit too... processed, somehow, too electronic-sounding. (Though the part where it mimicked the narrator was hilarious.)