I like thinking elements such as puzzles (as long as they're not too difficult and serve some logical purpose, preferably with multiple solutions) and limited weapon capacity, but I like to have a little more than that. I like a plot riddled with questions, shrouded in mystery, and dark enough to give you nightmares (without being survival horror. A game can be dark without zombies and demons).
Of course, not all games should be deep. Sometimes it's good to pick up a game like Ratchet and Clank with a buttload of weapons, one deminsional characters, and enough cracks at pop culture to make milk come out your nose.
Note: My sudden obbsession with dark and scary things has nothing to do with me running around the Shalebridge Cradle on Deadly Shadows. Really. "
EDIT: The only real problem with limited weapons is that if you know you'll need three guns in an upcoming battle, and you can only carry two, you have to run all the way back for the one you left behind as soon as you finish with one. This always happened to me on "Assault on the Control Room" on Halo. There was one area I needed a sniper rifle to take down some gunners on cliffs, a rocket launcher to deal with a tank, and an assault rifle to finish the job. I had to finish sniping as many things as I could, run all the way across the tundra, grab the assault rifle I stashed, and run all the way back. Then afterwards, I still had to find a pistol to replace the rocket launcher (I hate rocket launchers, too messy).