What is "depth" in games?

This has led to quite a discussion on BGG, so it's worth trying here:

What is "depth" in games?

I searched for “strategic depth” on Boardgamegeek and found surprisingly little in the way of definition. Lots of people used the term (or just “depth”), but they didn’t explain what they meant. It’s another of those “everyone knows what it means but does not define it”. (If there have been discussions of depth that my search missed, perhaps someone can point me to them.)

I have since found that the word strategic can be a cause for much angst, so perhaps we should just talk about “depth”. What gives games depth (or what makes games “shallow”)? Which games have depth? I think most people would agree that chess and go are deep games. And that Candyland and Tic-Tac-Toe (Noughts and Crosses) are not. What about Monopoly? Depth apparently has something to do with the complexity of decision making.
Perhaps it has something to do with the number or type of choices presented to a player. If you need a jumpstart you might look at “How Many Choices are Too Many” http://gamasutra.com/blogs/LewisPulsipher/20111025/8731/How_... which includes many comments as well.

So what is “depth”?