I haven't read much beside Batman, so I can't speak for every comic, but it seems to me that most comics are all about the hero vs. villain conflict, on some plane or other. Anything else is stacked on top of this central fight. I would personally love stories that focused more on the characters - for example, Batman & Robin Annual had very few fights, but it was so heartwarming and just plain awesome. I also wish for some more funny stories, and I do think there's a place for them.
In my totally unfounded opinion, it's the focus on combat that keeps the medium stuck with the teenager+ crowd. Which is fine in and of itself, but it would appeal to other population segments more if they went outside this format.
I believe many independent comics are breaking the mold and going other ways? At least I've seen some in the library that didn't at all fit the DC/Marvel mold, I just can't name any, as I'm a character (read: Batman) fan first and foremost.