NetHack. Downloadable for every platform ever at http://www.nethack.org/

Play it. Learn it. Love it. Die a lot.

Ah ... here's a good site for some old stuff.

Abandonware: http://www.xtcabandonware.com/index.php?l=news

Lots of good stuff on that site.

F.E.A.R. Combat (free multiplayer component of the popular FPS)
Return to Castle Wolfenstein Enemy Territory: a free mod of RtCW; beware, only very good players still play it though so it's hard to get into

X-Com: UFO Defense: also known as UFO: Enemy Unknown; it's abandonware - and it's one of the best PC games of all time.

X-Com: Terror from the Deep: essentially a more difficult, slightly modified clone of UFO defense; it should be abandonware, but I'm not sure with it being available on Steam. I won't link to a download, but google is your friend.

You may need dosbox to run the above two games.

Warsow: I saw this free FPS in the gentoo repositories, and it runs on windows and OS X as well. I haven't played it yet, but apparently it's pretty good.

Wormux: A free Worms clone initially designed for linux, but it runs on windows as well.

Wesnoth: A strategy RPG; I haven't played very much of it, but it seems decent.

Thanks for that abandonware site!  A few from it:

Warcraft: Orcs & Humans: First installment in the warcraft franchise!

Civilization: First installment in the legendary Civilization franchise!

If you use google you can probably find more. There's a multitude of free games out there.

I have found a great site:
http://www.the-underdogs.info/

What class (indie or big budget) and genre are you interested in? Because It's true there are probably very few big budget games that are free. But if you get into indie gaming it is a free goldmine. I play mostly indie games these days and there are a ton of great games. I know I've seen it here before but one of the best offerings is Cave Story. Battle for Wesnoth is a good strategy game. Stranded 2 is a wonderful sim. Plenty of great indie offerings out there. www.tigsource.com exposes many of the best.

Budget isn't an issue. A big budget doesn't mean a good game(Daikatana anyone?). I will also give any genre a blast, although I'm not to keen on RT/TB strategy.

 if you look at my list, a good MMO is LastChaos.

Scorched 3D: simple (to start, but you'll have to learn to approximate mentally ballistics calculations, taking account of the lie of the land and direction and strenght of the win too) and addictive.
It's a totally free improved 3D version of the old 2D shareware Scorched Earth.
It's multiplatform too, you can run it on Linux, Windows, OSX, *BSD, Solaris and every Posix compliant OS.
http://sourceforge.net/projects/scorched3d/