I found this article about window-box gardens. It mentions (among other things) that some of the easiest vegetables to grow are lettuce, radishes, beetroot, rainbow chard and rosemary. At our last apartment, I grew basil on the back porch - one pot provided me with a winter's supply. And I use basil a lot; it's one I put in nearly everything.
On the balcony - the lettuce & collards are still happy, and the cabbage is just starting to form heads. I haven't pulled the beets yet but I've given up on them - still just a leaf or two, and 3" tall. I don't think they like southern summers :) I have one happy branch of mint. I really need to re-pot it, and give it some fresh dirt. Hopefully that will perk it back up.
I've also got to go wandering through seed catalogs, and see what I can find that I can grow for winter. Swiss chard, radishes, turnips, perhaps another round of cabbage or greens, definitely lettuce. Remember, when planting a balcony garden. one can always buy sampler packets of seed. I did this when I could, and will get a few plantings from each packet.