Foods You Can Grow From Kitchen Scraps
These plants can be propagated as many times as you wish and all you need to get started is some left over kitchen scraps. You can start most of them with conventional, commercially grown plants. But sometimes commercial plants have been treated in ways to hinder replanting, so it's a good idea for your original piece to be organic. Of course, the rules of growing plants apply - put them in the sun, water them, don't plant in the garden bed if the weather isn't what the plant wants, etc. Y'know, things that should be obvious.
Fennel, Green Onions & Leeks
The same method is used for these 3. Stick the white root end in a glass with some water, leaving the cut end above the waterline. As the green part grows back, you can either cut it off with some scissors when you want to harvest (it will continue to grow back) or you can plant it in the garden. Change the water about once a week.
Lemongrass is done in the same method except you don't leave it in the water, you directly plant it after new growth starts.
Use the root end of an onion with 1-2 inches of onion "meat" still attached. Place the onion bottom in dirt. Cover it with 1-2 inches of soil. Once the new growth has developed remove the bottom from the dirt. If there is more than one bulb, separate them into individuals, leaving some roots on each. Plant in soil and cut off two-thirds of each shoot. The cutting will encourage the bulb to grow. Harvest as green onions or allow them to fully develop.
Cabbage, Celery, Bok Choy, Romaine Lettuce, Other Lettuces
Place the root end in a shallow bowl of the water with the top part just above the water line. Optional - occasionally spray the top with water to keep it moist. Change the water every couple days. After about a week you should have new leaf and root growth. At this point bury the whole thing in soil save only the new growth; keep the leaves above the dirt.
Plant a single clove in soil. Once the garlic has established itself again it will start to produce scapes (the green stalk). A common practice is to cut off the scape at the first leaf joint after it's made a few loops. The idea being that the garlic, having no flower to divert resources, will then focus its energy on becoming a full bulb. Remove a clove or two from the full bulb when you harvest and repeat.
Simply place a chunk of ginger root in soil with the newest buds facing up and it will start to grow anew. When you harvest, pull the whole root, remove a piece of it, and plant it again to keep the cycle alive.
You can regrow from any old potato that has grown large 'eyes'. Cut the potato into small chunks about 2 inches square, making sure each chunk has at least one eye on it. Let the chunks dry at room temperature for a few days, this will keep the potatoes from rotting after planting. Place each chunk a foot apart along an 8 inch deep trench with the eye(s) facing up. Build up the soil around the plants as they grow, this will allow room for more tubers to form. This is the standard method for propagating potatoes, so there are many different methods for how one can plant the chunks.
You'll want to use organic, because the majority of commercially grown varieties have been sprayed to discourage them from shooting. There are two methods to produce "slips" (the name for the shoots used to propagate sweet potatoes). Either partially or completely bury a sweet potato in dirt or partially submerge half a sweet potato in a cup of water (supported out of the cup with some sort of skewer.) If you used dirt, don't forget to water. When the slips have grown to about 4 inches, simply pluck them off from the base, plant them, and watch them grow.
Remove all the fruit "meat" from the crown of a pineapple. Cautiously continue to cut thin slices off the end of the crown until you can see the root buds (a ring of little dots around the outside of the pineapple's base). Remove a few outer layers of leaves leaving around an inch's worth of base. Directly plant the crown in soil or submerge the base in water until roots develop and then plant. If growth is successful it will take 2-3 years before the your pineapple plant fruits. But the wait will be worth it because home-grown pineapples.