1. Brainbread. Half-Life mod (not 2, the original) that has you and your friends fight a metric ton of zombies at every corner. Has objectives, but they aren't VITAL to play the game. Oh, and there are three skill trees that you can upgrade too along the way; Speed, Skill, and Health, I think.
2. NMRiH HL2 mod, and it's *really* hard, in my opinion. Dunno anyone who's beaten it.
3. Obsidian Conflict - Best Co-op mod for the HL2 engine; I recommend having CS:S and DOD:S mounted on it (oh, and Ep1 and Ep2), while it's not free, I expect you'd probably be able to play it.
4. Planetside 2, requires a high-end PC (I think so, at least), but it's a play-to-win game. However, you can still play w/o paying, I do, at least.
5. Empires mod. It's really dead, but it's basically 1 Commander who is RTS, and then the army, FPS. Two teams fight it out to blow up each others bases. I had a blast with it, when SKG had a server, I'd recommend playing it if you can find a server.
I'll post a few more when I get home, tomorrow.