I tried out some Muds a while back...there was one where a few classes could get xp without fighting. It was a lot of fun at first to play a necromancer stalking high-level parties to steal the corpses of their kills
But in the end sitting around for ages waiting for a crafting timer to reach zero is just boring...and I'm not sure how you could do non-combat stuff without something like that.
Of course, combat can be like this too depending on the game. I've heard good things about Final Fantasy XI's combat though.
Not that I've been playing much MMoRPGs lately.
How about this?
http://ukosite.jolt.co.uk/imprunner.html
Peaceful competition to beat other people's highscores
I had a lot of fun with it anyway...more than some games I've paid for.