Puzzle Pirates is original, but I never really got into it. I hope Star Wars: The Old Republic does a good job with the story, but they're definitely planning to have fighting be central to the game. Free Realms and Endless Forest don't have a compelling plot or interesting goals. The Sims Online failed, and A Tale in the Desert is mostly about crafting. I think crafting is cool, but it tends to rely on gathering. And gathering resources bores me to tears.
Matthew Everett posted a great analysis of why violence is a convenient mechanism for game design. Dana Massey posted a wonderful suggestion for a non-violent mystery MMO, which I find fascinating. If somebody created a good mystery/adventure MMO, I know one thing for sure.
I'd play it.
Captain Dwur posted a list of ideas for genres that can provide non-violent RPG challenges: adventure, detective, natural disaster, political, and puzzle. Some of his readers posted other ideas, such as card games, treasure-hunting, getting close to animals to study them, or competitions based partly on numerical skills and partly on strategy (e. g. footraces, archery, art/poetry). Syp also wrote a commentary on some common non-combat elements of MMORPGs.
The new Mistborn Adventure Game is a tabletop RPG that focuses much more on mysteries than combat. When questioning someone, the GM has a list of "secrets" that are revealed in order. If the person resists questioning, there are three ways to challenge them: physical, social, and mental. The victim is "unconscious at 0 Health, disgraced at 0 Reputation, or devastated at 0 Willpower." (See Crafty Games's downloads section for a free intro to the game.)