Stuff like this tickles my fancy, though I have yet to employ most of these I did create a list of challenges tailored for FPS/TPS games or just action games in general. These are for those who think playing on the highest possible difficulty just isn't good enough.
No HUD: Disable the HUD if possible. If the game has sound and/or visual cues of low health/ammo this is actually not that hard.
Coop Mode: Play solo in enemy enriched co-op mode if possible. This one only seems to matter for Doom engine games, there's often bosses in unusual places and just a lot more monsters around in general.
Keyboard Only: No mouse allowed. This can be difficult if it's games that were designed with keyboard + mouse. Default or custom controls is up to you, but it has to be strictly keyboard.
No Autorun: Manual running. Newer games have limited sprint which doesn't really count, but I won't let that stop you from trying to play without it. Basically you have to hold down an assigned key in order to move quickly all the time.
No Health Items: Scorning health pickups. Consider yourself a pro? Try working with the only health you have. Can be on a by-level basis, but you absolutely cannot use health items.
No Armor Items: Neglecting armor pickups. Ditto. You can have all the health you want, but no protection to decrease damage taken.
No Power Ups: Avoiding power-ups. General pickups are fine, but any temporary super-boosters are zilch.
No Inventory Items: Ignoring inventory items. Pretend the inventory system doesn't even exist, good for BUILD engine games.
No Upgrades: Stock/underpowered character playthrough. Can be very challenging depending on the game, perhaps even downright impossible near the end.
No Damage: No damage received either via level based or the entire game. As it says on the tin, if you get hit once you already lost this challenge. Hitscanners will be your worst nightmare.
Stock Start: No consecutive play per level. Commonly referred to as the 'pistol start', the weapon you begin with in question varies game to game.
Hub Start: Ditto for hub based games. Quake II refers to its hubs as 'units' but it counts all the same. Quake II actually allows you to execute a config in the console to begin a particular unit and gives you the appropriate weapons and items for a fair start. That's too easy, so manually warp to the beginning of a hub instead.
Start Saves: Only saves at the beginning of a level. Dying means restarting, but you can keep the stuff you got from prior levels.
Hub Saves: Only saves at the beginning of a hub. Dying here will be a problem as it means a lot of work lost.
Progress Saves: Only saves when acquiring important items or performing important tasks. This is one of my favorites in key-based FPSes, I would save the game only when I got a key. The number of keys dictates the number of possible saves per level.
No Saves: No saving permitted. If you lose its back to the beginning with stock equipment, kind of like an endurance run, except you can repeatedly fail on this.
No Secrets: No aid via secret areas. Not all games require mandatory secrets to finish but some will be notably harder, maybe impossible without the aid of certain secrets.
Shareware Weapons: Only use weapons found in the shareware/demo version. For example on Doom you would not have a plasma rifle or BFG. Quake 1 isn't as bad since it only added the thunderbolt in the registered version. Other games could be harder.
Melee Weapons: Only use melee weapons. No firearms allowed. Might not be doable on all shooters.
Bullet Weapons: Only use hitscan weapons. No projectiles allowed. Might not be doable on all shooters.
Weaker Weapons: Only use lower-tier weapons of superior ones. For example a shotgun vs. a super shotgun.
No Special Weapons: No exceptionally powerful or unique weapons. Be it the last weapon you acquire, something with unusual properties, or extreme damage.
Start Weapons: Only use what you begin with. Again might not be possible on all shooters.
Original Weapons: Only use base game weapons, no expansion content. Self-explanatory.
First Person Mode: Play strictly in first person if an option. If it's a third person game by default this can radically change the way it plays.
Third Person Mode: Play strictly in third person if an option. If it's a first person game by default this can radically change the way it plays.
Fast Monsters: Use fast enemies parameter if available. This only applies to old id games sadly. Enemies are twice as fast and aggressive.
Respawning Monsters: Use respawning enemies parameter if available. Again an old id games stable. Enemies don't stay dead and continue to hound you throughout a level. Hexen II actually has a secret fifth difficulty that makes monsters (and mana pickups) respawn but its only in the expansion.
Endurance Run: No saves whatsoever in the entire campaign. This is the ultimate do-or-die challenge, it effectively means playing the entire game in one sitting and never failing once. If you die, it really is game over. You used to have to play rougelikes for perma-death, now you can have one anywhere!