My 16 birds are in an 8X16 coop/run so they have an OK amount of room. They seem to keep pretty busy doing various "chicken activities" like scratching, basking in the sun, doing stuff with each other from just hanging out, some chasing etc. (Got a couple of grown roosters in there and a couple more coming up.) They also get to watch me do stuff, I'm always moving stuff around, doing stuff, so I'm sure I'm pretty entertaining. Then they get veggies, and random treats at random times. I have a lot of walnuts I've gathered and those are good, they have to peck 'em a bit to break 'em down, I keep an oatmeal can of shelled ones and they get a handful or two of those. 
Cabbage (or other veggie) tetherball sounds like a great idea.