Even though am listing only my favorite ones.
World of Padman
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Guess what? This game was made using Quake 3 engine. Just Imagine Quake 3 without terrifying looking monsters and the game support bots too, BTW the weapon isn't so scary looking but consider a toy-ish version of Quake 3's.
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So if you don't want your friends to laugh at you just because they fragged you the entire hour, so this is your escape and if you practice harder, might have a chance on a come back if you've been trough that. :) Now, don't waste your time, go here: http://www.worldofpadman.com/ and download it now!
If you want more, there's a list of free FPS at Wikipedia you might want to check out. Happy fragging!