All experimental for me at the moment, but, I do a deep litter in the small run under my coop/bedding box. The upstairs bedding area has a fairly normal layer of bedding material which is kept as clean as possible and DRY. In my outdoor chicken yard, I do a compost system - Jason Rhodes style but scaled a little for my suburban backyard.
The inside run is on concrete so I put a layer of good soil to bring in necessary good guys to make the system work.
I wanted to try this without too much expenditure (because I was not confident it would even work) so I used homemade lemon grass mulch and grass clippings (fairly dry - ie mowed when lawn low and dry) to bring up to about 6 inches (much higher than that and the hens have no headroom under the bedding box and the ramp up.) A low board across the doorway keeps the depth inside instead of it being spread across the doorway and out of the coop by all the scratching and digging activities. The number of evenings I had to get my spade and clear the doorway before I could shut the coop up, the lazy chicken-lover in me was happy to see the difference a little bit of timber made.
If the hens don't dig it all over during the day, I rake it every evening, to make sure all dropping are turned into the top layer, and I add more bedding or clippings as needed. I also throw in a shovel or two of soil, which is full of life, to encourage them to do the work of turning the soiled top layer in as well as to provide the joy of dirt baths and foraging for worms and bugs.
It ain't a great way to make compost as far as I can tell (though I haven't got the depth to do a proper whatever-it-is-I'm-doing, anyway, but the hens love to dig through it, lay in it and find yummy critters that might be hiding there and I have yet to clean the inside run out (after 2 1/2 months) and it doesn't look like I will need to for some time yet.
Although under roof, the whole lot gets some driving rain occasionally and a bit of run-off across the concrete, too. Still stays overall dry and no sign of mould or bad smells. I never put food scraps down in the inside run, though I do occasionally throw in various healthy and sweet smelling herbs (Lisa Steele style) and maybe a handful of seeds or grains to encourage scratching.
I don't throw their food on the ground in the outside run, either, though they sometimes make a mess, which I pick up and throw in the in-yard compost. They usually jump straight in there and clean it up, anyway.
I'm pretty happy with all three areas and basically the only money I am spending is on the nesting and sleeping areas where I use commercial absorbent bedding as well as some homemade lemon grass bedding (that stuff smells divine). It is a bit of work keeping up to the actual sleeping quarters, but I would rather that area be as clean and comfy as possible as they can't just move away if they don't want to be there.