I played Pokémon Heartgold for 30 hours (beat the first 8 gyms in that time, despite playing glacially slowly) without using any items except Balls, without having a Pokémon faint, and without saving except to suspend the game.
Finally a friend of mine told me "that's fucking stupid", and snapped me out of it.
I played through Darkest of Days with only pistols, until I loaned the game to a friend who hasn't given it back yet.
"White Whale" challenges that I completed would include getting these achievements legit:
Stomach Upset (Left 4 Dead 1, go through a whole campaign without a Boomer puking on anyone)
Do Not Disturb (Left 4 Dead, go through a whole campaign without anyone startling any Witches)
Zombie Genocidest (Dead Rising, kill 53,594 zombies in one playthrough. I died once at about 45,000 kills, so I had to start over completely)
7 Day Survivor (Dead Rising, last 14 hours in Infinity Mode without saving. In Infinity Mode, your character starves to death after 20 minutes without food.)
Guardin' Gnome (Left 4 Dead 2,
I have 53,000 kills in Left 4 Dead, and there's an achievement for 53,595 (take that, Dead Rising!), but I'm not playing for a little while. I want to get the achievement for surviving a campaign on Expert difficulty in the same playthrough as the achievement for 53,595 kills.