Joy of Cooking says to cook the chicken with a mirepoix (diced carrots, onions, and celery) and a bouquet garni (a fresh or dried herb bundle- see, if you had the Food Lover's Companion, you'd know these things!). But Jamie Oliver says (and that's one reason I like him) just throw in pretty much whatever you have on hand that's savory.
I covered my carcass with cold water and added fresh thyme, oregano, and rosemary, 3 cloves of unpeeled garlic, half a roughly chopped onion, and a chopped carrot (ain't got no celery). I brought everything to a boil, then reduced it and let it simmer for a few hours. About halfway through, I added another 2 cups of cold water, since the broth had reduced. You can also add bay leaves, marjoram, whole peppercorns, parsley, leeks, cloves... if you'd add it to soup, you can put it in your stock!
After I strained it, I was left with this!
I have it in Mason jars to cool down and then skim the fat off, but I'll freeze some of it in ice cube trays (I thought this was a particularly ingenious idea, since you can then use as little as you want without thawing a whole thing of it).
So are you going to try it? Cuz you should!