The bread that Dave suggested definitely works -- I've done it before.
If your cookies have a lot of brown sugar or molasses, they won't dry out very easily, but it's still good to cover them if you can.
If you make a recipe with a low amount of sugar, it can go bad more quickly, too. I used to make these yogurt-oatmeal bars with with only 1/4 cup sugar, and they would start going bad in a couple days if I didn't refrigerate them. Regular chocolate chip cookies, on the other hand, last longer, but they've never stuck around long enough to really test them, haha.