I normally associate "deep" with "challenging".
I feel like a lot of the time when ministries or Bible studies only scratch the surface on Scripture/issues/etc they get accused of not being deep enough... but usually the people saying that are either A. not participating and digging in or B. genuinely not being challenged.
I don't know if that makes sense, but I figured I'd throw my 2 cents in :)
I think most people associate "deep" with "I never thought of that before." The truth is that the meat of Christianity isn't hard to understand: Love the Lord, serve your neighbor.
The perfect Bible study would help you do those two things. There's not any more too it, but for some reason people often are looking for more than that.
The way I understand it is people want more like "academic" Bible trivia-esque head knowledge. I guess like more than what you typically hear, as in the "go-to" type verses for a given situation. To me it comes across as they want more John Calvin/Piper, less John 3:16. And throw a little Romans 9 in there. If that makes any sense.
Also I think there are passages that tend to be discussed over and over again (sermon on the mount, etc.), while there are other passages that hardly ever seem to be discussed in a small group/Sunday worship setting (Romans 9, for example). I think there's a reason for that though. Small group material should be more relational/applicable than "deeply theological". That's why people go to small groups...the encouragement and nourishment associated with community and fellowship with other believers. If you want to dive in and learn all the intricacies of complex theological ideas, I say go to seminary or read a bunch of books or something.
To use an analogy, I think people are wanting more steak and less milk. But I think some people want that 72 oz. steak that you can get for free if you can eat it within an hour, and in the end they might realize that maybe it isn't what's best in light of the big picture. They probably can't eat that much steak, even though they were pretty hungry to start. And in the end they'll have to pay for it since they didn't finish...and it's a lot of dough to account for. I admire their longing for understanding, but think they might be going about it the wrong way. I used to be like that, even.
I dunno about the whole being challenged thing...I've read James a billion times, and it wasn't until this semester that it absolutely pounded me into submission. I was quite challenged (huge understatement), despite the fact I was already pretty familiar with the book to begin with. Sometimes you can read a passage you've read like 50 times and something jumps out of nowhere and you see something you've never seen before. I'd challenge the people who claim that the words of God are no longer deep enough for them to pray that God would give them the eyes to see what he wants them to learn, rather than complain that reading about the prodigal son for the 100th time is boring and not "deep enough". Pray Pslam 139:23-24 for starters. It will get your tail swiftly kicked, in due time.
/rant
sorry :)
I agree 100% with Jared.
As for a ministry itself, I think the fruit of a deep ministry is when those involved begin to truly love each other and the people outside. I've known a lot of theology nerds who are jerks (and I've been one). The deepest people I know are those who live the gospel, regardless of whether or not they know what penal substitution means.
When I think deep I think "meaty", as in more than milk. Something that applies to your everyday life and makes you stop and think.
I think for many it just means "intellectual" or "theological."
I think Jesus is depth.
I've written about it here and here. Fwiw.