What do you mean by depth?
IMHO, character depth comes from his/her philosophies, motivations, goals, fears, etc. and how he/she applies those (reacts) to the events that are transpiring around him/her.
For eample, my current character's goal is to become a deity. Short of that he is willing to consider lichdom as a possibility. He fears death and the eternal servitude that comes with it. In general he is cautious and calculated but will take risks if he feels that it advances his over-arching goal. He acts aloof in his dealings with people but secretly notices everything and takes great pains to right any percieved wrongs. He is a stranger in the country he is adventuring in.
IME this character has depth - and yet he adventures in a sandbox (one created before I made this character). I don't know if he will succeed or fail - to me that is part of the fun.
Ahh, now we're getting somewhere.
I agree, this character has depth. He has goals and a personality. I might argue that he lacks a bit of relationship with the other characters and the people in the setting, but, that's a quibble and not terribly important.
Now, how will he succeed at this goal? What in this setting will allow him to achieve this goal? How, specifically, will this particular setting allow him to achieve this goal? Other than the standard D&D thing of simply killing your way up the levels until you reach godhood or lichdom, which can be done in any setting.
This is what I'm talking about. To me, another way would be to have a setting where your character is on his road to godhood
of a specific people. You would have ties to those people, links, history, a background stretching back. Your achievement of godhood would likely be pretty much guaranteed, so long as you continue playing.
However, the campaign wouldn't be about that specific goal. The campaign would be about building your following. What kind of god would you be? How does becoming immortal affect you?
Look at the new Dr Who series. Many (if not most) of the episodes revolve around the Doctor's incredible loneliness. The fact that he's the last Timelord and nothing he does can change the fact that he's alone, for all time. It's an incredibly sad story.
In the "I'm gonna be a God" story, we would work together to lay out some themes and concepts that should be brought up during the game. The game would be about the exploration of those themes.
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Another weakness of this approach is it really doesn't work well with large groups. You'll never get a group of six or seven people to be interested in such a limited storyline. It really won't work for any length of time. However, in smaller groups, particularly in smaller groups with similar interests and playstyles, this can work very well.
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The Ghost - I'm not in any way saying what you are doing is wrong. There's been a few people here who are trying to paint this as an either/or sort of thing and it's not. I'm simply bringing up alternatives.