Well, Portal 1 and 2 fit that description perfectly (both great), but you probably already knew about those. Hmm...
Phoenix Wright on the Nintendo DS (also on GBA, Wii, iOS) fits your description somewhat. It's basically a menu based adventure game but instead of absurd adventure game logic you use real logic to figure out the contradictions in the witness testimonies, present evidence and so forth. The investigation parts can be a bit slow, but the characterization and humor make up for it. Courtroom trials are where the real fun is, and there's plenty of action to be had there and when the case takes a dramatic turn in your favor it feels pretty awesome.
It kind of also depends on how strict you are with the requirements of the game being absorbing and having little to no combat. For example, Psychonauts has some combat and it's a platformer, but majority of it is against figments of people's imagination, thus you could say it's mostly nonviolent. There's plenty of wonderful things in this game too, like the music, stylized graphics, level design, originality, characters and dialogue. Yahtzee's review: http://www.escapistmagazine.com/videos/view/zero-punctuation/2-Psychonauts Best played on PC with a gamepad, and if you have an AMD GPU you need to look for a workaround for a FPS drop later in the game, the solution can be found on Steam forums.