Well, since we're being vague...
I'll just tell about different games with different sorts of depth then.
Story depth:
Planescape torment was in my opinion extremely deep story wise. The Nameless one, the character you play as, is an immortal. This isn't something you realize through dying in-game, you realize it the first moment of the game. You wake up in the mortuary, with a talking skull as your companion. Throughout the game you realize that The Nameless one has lived countless lives. This gives the game a lot of "story" depth, because things usually aren't what they initially seem to be, and somehow the Nameless one is usually involved.
Just play the game, it's extremely hard to tell too much without spoiling some seriously awesome moments you'll have in gaming. Prepare for a person that has lived an awesome life, and a story that has some surprising turns and revelations. If you want to get the most out of the story, which is the great thing about this game, make sure you get a high Wisdom and Intelligence stat.
Philosophically deep:
Star Wars: Knights of the Old Republic. TWO. Damn. If other Star Wars games or movies weren't as deep you'd expect from a franchise that deals with mind boggling stuff as "The force", try this game. If you've played Kotor 1, you'll already know that the Knights of the Old Republic takes place in...The Old Republic (Duh). Kotor 1 is all about beating the bad guys, or being the bad guy and winning it all. Yay!
The secound installation however, was something quite different, which might have put a lot of previous Kotor fans off. By the time i had finished it for the first time i was pretty confused. I didn't know what had happened, what was the deal with one of the main "villains" and why things turned out the way they did. I didn't even fully understand the character you had played the game as, known as The exile (Or whatever you named him/her really. Shug Repner was my exile!).
After a second, and a third, play through though I was starting to patch things together, and finally I could understand the cryptic Kreia, also an exile of sorts that joined your party in the game. Still though, a lot of stuff is really up to you to figure out, and some stuff you just have to guess at. The Exile will to me always be a mystery of sorts, but I have my own philosophy on what his deal is (As everyone else seems to have their own in the game, which is really confusing).
The game isn't quite as obvious as Kotor 1, where you have your main villain from the start. "Ah, there's the bad guy. Good, now I know who to kill."
The villains of Kotor 2, or mainly one, are however not as obvious, or have as obvious plans as the main one in Kotor 1.
If someone could instruct me how to put a "spoiler" tab, please do, so I could spill my guts. I can't go on like this!
Anyways, Kotor 2 is really philosophically deep. I've spent hours discussing the plot with friends, it's a truly deep game that has so many levels to it. Try it, but prepare to be confused! And if you're not, you're not looking deep enough
Just as a mention, Torment could easily fit with the philosophical category as well, and Kotor 2 with story. Not going to go on like this forever though, let's switch category.
Game play depth:
Strategy games in general confer a lot of depth, especially when we're talking online play. Games where you can really play the game in a LOT of different ways and come out on top, this is what i consider to be real game play depth. Multiple races, which is the standard nowadays I guess, makes things even more deep.
Play DotA, Defense of the Ancients, for serious depth and fun times!
Shooters really don't offer too much depth in my opinion. Not that it's more boring because of it, but usually there aren't too many layers of how you can do things. Star Wars: Jedi Knight: Jedi Academy however offers a lot of different ways to play things, especially if you're playing Siege. Lots of depth in that game!
A lot of Final Fantasy games are really game play deep, the one that springs to mind is FF7. Chrono Trigger with their Dual and Triple techs make things interesting as well.
In general, I think games that offer an online component right there gives a lot of depth, because people are always more unpredictable then a computers programing, which gives you the ability to use different tactics. Offline games can offer depth though with an intricate leveling system, like Arcanum or Fallout. Fighting games can also have a very deep way of being able to beat the other person. There's usually a lot of different ways to fight. Damn i need to play more fighting games though, last one i played was Tekken 3 at a friends house for nostalgia's sake.
Depth of...uh...graphics?
Yeeah graphics can be deep...like...yeah...
Time for me to stop bullshitting you guys. Out!
(Mario! It has red colors and blue colors and round shapes, square shapes. Now that's DEEP)