My vote is for HL2: Episode 1's 'The One Free Bullet'. You only get to fire one bullet throughout the entire game, to knock off a padlock (Rockets, crossbow bolts, grenades, energy balls, physics objects, etc. are fine though). Runner up is 'Little Rocket Man', from HL2: Episode 2, where you must carry a garden gnome through the entire game and deposit it in the missile at the end.
So, what are your candidates? (Note: Grind achievements like dealing one million damage points and so on don't count, because they're not really hard, they just take an ungodly amount of time.)