Not killing stuff

[loramin]

You can level all the way to 60 in EverQuest without doing even a single point of damage. Obviously this is a lot easier for healers, but technically it's not impossible even for a Warrior ... it's just insanely difficult (you'd have to complete low-level quests an insane number of times and fish/forage/tradeskill/track/loot decaying corpses/etc. so you can trade with players in exchange for gear and stuff that you can turn-in for XP).

Although they're not online you might want to check out old school adventure games: they have all the puzzle solving, questing, etc. that RPGs have, but without combat (and usually without death too). But if you want to play in an online world with other players I'd check out Ikariam. It has a tiny amount of fighting, but it's mostly about trading/building.

Of course there's always a million online puzzle/trivia/etc. games with no fighting, but those might qualify as "kiddie games".