Well, I am a big fan of you people's website (already have 2 games) and am wondering if there are any games that can be modded to the fullest. I already have Sacrifice and The Guild, and although Sacrifice can be modded a lot, there wasn't as much as I wanted (need... more... mods.... XD) Anyway: I would like a game that has an excellent community that is still ongoing or already has lots and lots of mods. If you tell me that the multiplayer community in one is a problem, then please make sure the game has many mods that are singleplayer (bots count). Bonus points if you tell me how good multiplayer is in the vanilla (original) game too! Thank you guys for the excellent community! :)

I'd go with UT2k4 for that. Mods aplenty and still going.
As for the vanilla multiplayer, it's still going strong and good and fun :)

I'm sorry, I forgot to mention that I had a full pack of Unreal Tournament games except for 2003. I also have Half Life 1 & 2 and many other source titles.

each GOG game has it's own forum on the site, and mod-friendly games get a sticky with a link to good mods in their forum.
I actually came here to link you to the general forum mod thread but I can't find it. It's around here somewhere...

Sadly, UT2k3 was quickly overshadowed by UT2k4, so I wouldn't worry. But it is, as Andrew said, the most moddable game on GOG.
There are a lot of mods for many games here, in the cases of the Fallout series, Arcanum et cetera, they're mostly fan updates to replace content that never made it into release, bug fixes and so forth.
If you want total conversions, UT2k4, as stated, Duke 3D, the Freespace series, Ghost Recon.. And yeah, check out [url=www.moddb.com]ModDb[/url] to see if there are mods for a game you might be thinking of purchasing.
UT2k4 vanilla is awesome in multiplayer, sadly there aren't many Australian servers still up so I don't play it much, but when I had, it was a blast.

Freespace, Duke, Arcanum, Creatures, Gothic...

Giants still has a fairly active mod community... well, maybe not active, but the good mods are still maintained.

FALLOUT, FALLOUT 2

Freespace 2. Grab Freespace Open to go with it and there's a massive amount of mods available.

In a way you have 2003 as 2004 has all the content from that game.

Yeah, 2003 was basically the (thoroughly tested) alpha version of 2004, which is why it has been largely ignored by most people ever since.

Duke 3D if singleplayer's your thing. There are tons of singleplayer maps and mods out there.

You just made me buy blood.

Have fun with it! I know I do have.