World of Warcraft, several players have tried their hands at different methods. Gutrot from the EU leveled to 70 wearing only a loincloth. Gweryc decided to run his hunter as a melee-only character. Noor the Pacifist then decided to level his character to 70 by not killing any mobs at all, and he only did quests that involved non-violent criteria (like FedEx quests).
Now we hear that another player has upped the alternative-leveling ante by not questing at all. Everbloom, level 85 Night Elf Restoration Druid from the Feathermoon server, leveled all the way to 85 by grinding exploration and gathering experience! While Everbloom's armory profile shows that the character entered one five-man dungeon at some point, there is only one completed quest -- and Everbloom explained that one was a fluke. Apparently Blizzard blindsided her by granting a quest completion for simply reading an Argent Tournament invitation letter -- even without having a quest for it. Everbloom additionally managed to level to 85 while only dying 25 times in total (once to falling), making this crazy stunt an even more impressive feat.
If you'd like to read more about it directly from the source, head over to the World of Warcraft forums and check out Feathermoon's leveling thread. It's certainly worth a look if you've ever thought about trying some form of self-imposed limitation leveling in your MMO of choice.