Kale! Chicken and kale soup is one of my favorites! (I also stick some chopped tomatoes, onion, and garlic in there, also sometimes Spanish or Portuguese style chorizo.) A Mexican style chicken soup, with white rice if you eat it, and jalapeno, avocado and onion is also good.
I also use it for most vegetables I'm steaming, like broccoli, kale, or other greens. When I do eat white rice a few times a month, I always cook it in chicken broth instead of water.
I also use it when braising pork or potatoes.
Also, I'm Southern, so I adore collards, and they are Paleo-friendly. 3 parts chicken broth, 1 part water, a ham bone, onion, jalapeno and garlic (I put a jalapeno and garlic in about everything, but that's up to you) and let that simmer for about half an hour to get the broth good and smokey. Then add your well-washed and torn collards and cook til tender and no longer bitter.
I never have enough broth, myself! If I ever have any extra, I freeze it in ice cube trays, then pop out the cubes and put them in a labeled bag in the freezer so I can just pull out some cubes.