Anything that you put in the run will need to be protected for a while--even if they don't like to eat it, they *WILL* sample it, and they'll potentially uproot ANYTHING by their enthusiastic digging. Also, chicken poop is very "hot" as a fertilizer and can burn plants, so it would be well to keep them away anyhow.
That being said, I planted Black Hills spruce tree in the run, protected by a rebar cage lined with chicken wire. Black Hills spruce is a slow-growing spruce, but it gets big. I only planted it there because I am planting a windbreak through the area and the chicken coop will probably be rebuilt by the time the tree needs room to expand.
How much space to you have? You might plant a small tree that drops fruit in the fall--the chickens would LOVE that. My chickens enjoy eating the crabapples that fall from my crabapple tree (unfortunately the tree is far from the coop so I have to carry buckets of them to the chickens), and pears from my pear trees. There are PLENTY of semi-dwarf or dwarf fruit trees that might do well in your coop. What garden zone are you in, how much sun does the run get, and how much space do you have? Is the run to the south or north of the coop?
I agree they will sample EVERYTHING. 2 that I like are ornamental grasses. I have the white/green leaf one that grows about 5-6 feet tall.Fast to grow for cover. Also rose of sharon bushes. I have lots of seeds for that one.A bonus for that one is the flower buds attract jap beetles which I shake into a water bowl for the hens.The grass will give decent cover in one season.The ROS in 2-3.