1. Cassava. You'll need to find someone with cuttings. The cuttings root without a problem. I have started with pieces no larger than my thumb and, by the end of the season (Thanksgiving), they produced nice tubers. They are an easy, undemanding crop. The more care you take of them, the better they'll produce, but even without additional water or fertilizer, they'll still look good and produce something edible. Delicious boiled and mashed, or boiled and fried.
2. Malanga. You can get tubers for these Elephant-ear lookalikes from Publix or any market that caters to folks rom the Caribbean. Look for smallish ones that are heavy for their size. Stick 'em in the ground and wait. Like cassava, these plants are pretty indifferent to conditions--they are, for instance, the only tropical staple that can be grown in shade. Mine last year produced poorly--bugs got them. I think that if I had planted them somewhere with more sun, added some compost, and fertilized them, they would have produced well. This year I was pleased to see half a dozen little plants appear where I'd pulled the original plants. Much like other alocasias, they must produce lots of little corm-lettes. We'll see if they produce better this year with all the rain.
3. Okra. It does very well in large pots--better even than in the ground. Productive and tasty. I grow the burgundy available from Southern Exposure. The pods are edible even when they get large (six inches or longer). Not at all a fussy vegetable to cook--I slice it and fry it in olive oil and it's delicious. Deep-fried ain't half bad...
4. Hot peppers. I've always had great luck with Tabascos, jalapenos, and habaneros. They do best in pots for me, and require daily watering and a bit of shade in the afternoon. When grown in pots, they're perennial--my Tabasco is four or five years old and produces better with every year.
5. Caribbean seasoning peppers. There are several, and all look just like habaneros but are very mild. I grow a St. Lucia Yellow Seasoning pepper that's sweet and fruity and tasty. Prolific, too. Perennial in pots.
6. Sweet potatoes. Couldn't be an easier, tastier, more prolific crop. I save back small potatoes from the winter garden, leaving them all winter on the back gazebo, then plant them in some well-drained compost in spring and cut the slips as they form. You can stick the slips directly in the ground and cover with compost, but it's a better bet to pot them up in the shade for a week. Give them room and a bit of water now and again.
7. Roselle. I'm growing this mallow for the first time, in the same bed as the okra--they look great. Very pretty. Tasty drink made from its calyxes. I guess not technically a vegetable, but grown like one.
8. Peanuts. Go buy some raw green peanuts from Publix, stick them in the ground anytime during the summer. Forget about them. Dig them out in November or December. Enjoy. Really, that easy. They need nothing.
9. Okinawan spinach. Tasty, perennial, incredibly prolific, totally care-free. You'll need to find someone with cuttings... Good cooked or raw. High ornamental value.
10. Small (100s or smaller) cherry tomatoes. These can be started anytime. I grow Matt's Wild, which are very motile and something of a pest... But tasty enough and low-maintenance. Just make sure to keep it in bounds.
11. Cowpeas. I love my Mississippi Silver (from Southern Exposure). Totally undemanding, prolific and pretty. Very tasty. Belong in every garden.
12. Limas. This year I'm growing Willow Leaf Lima (from Southern Exposure). Planted in the spring sometime, it's just now coming into production, but it's a great plant--vigorous, twining, pretty, drought-tolerant and prolific. The beans are smallish but tasty.
13. Eggplants. They're perennial if you nurse them through the winter. I grow the "Little Fingers" which can be used small or left to grow large. (Interestingly, the "Little Fingers" have the largest leaf of any eggplant I've seen.) I love eggplant in Pasta Nora and as baba ganoush.

On the herb front: Basil (especially Thai), scallions (which I start from roots bought at the grocer), Mexican tarragon (pretty, too!), wild arugula, fennel (currently under serious attack by the swallowtail cats), epazote, pápalo (not for the faint of heart!), leaf celery (which I use as a replacement for parsley). Rosemary, mint, and thyme struggle, but generally survive the summer. Summer savory and certain oreganos (large leafed) thrive. I also grow chaya, which is OK but more of an ornamental than a food plant. The butterflies love it.

I'm sure there are other tropical crops to grow, but all of these crops are pretty easy. Many are perennial, and those that aren't can be started from saved seeds.