Thanks for the great feedback so far! I think the discussion shows that this is not an easy subject, and we can mean different things by saying that a story is "deep".
Genejoke & bravo1102: I agree with you that depth is clearly not required from a story. I think a story can be worth my while even if it isn't deep. But, apart from works I can enjoy with an ironic distance ("it's so bad that it's good"), I would require a story to be well told and somewhat smart. Which would imply that telling a story well is independent of how deep that story is.
Personally, I've found that I like a good mix. I need my simple adventure fiction to be complemented by something deep. Either in the way that one story combines both aspects, or it just means I have to alternate between deep and "shallow" stuff.
I think ayesinback has a point about fleshing out stories (and their characters), to make them more "full" and thus giving them more depth. It even makes sense metaphorically: The fuller something is, the more distance there is between its surface and its bottom (= depth). Or (again, metaphorically) it could mean increasing density, giving it more substance.
But I'd also say that that isn't all there is to depth. I agree with Banes: It is also about topic. And that's how I take ayesinback's suggestion to look at The Little Prince. The Little Prince is quite minimalistic in every aspect. But most fans of this book would probably say that it is full of deep insights into the nature of human life, how life should be, what's important in life, etc. Personally, I'm a bit sick of The Little Prince, because I feel it often gets abused by shallow people to decorate themselves with something deep. Its content isn't being discussed, it just gets quoted over and over, which is obviously the end of all deep thought. That isn't just the fault of the book, but also of the culture surrounding it. But in a way, the book invites this behavior, being written like a collection of quotes to begin with. Pretty much the same goes for Nietzsche and Wittgenstein, who are far too quotable for their own good.
Based on this feedback, I'd modify my account from my first post as follows. Depth can be/depend on:
- Topic (as bravo1102 has suggested, this might also depend on the audience, and what topic and presentation of the topic they're open for)
- Stories and characters being well-rounded, fleshed out, "full".
- Stories being layered: Its meaning being somewhere beneath the surface, possibly with different meanings being hidden underneath it.