I've thrown beans in the ground, they show up just fine. The germination rate may not be the 80% required by most state laws for seed sales, but they are as cheap as...beans. There is a high probability the beans have been treated, sprayed, fumigated, or otherwise chemically serviced. Using the storebought beans as your initial seed stock to develop more seeds can alleviate these worries. Beans start to drop their pollen before the flowers open-I've found the resulting crop to be true, with no hybrid issues. I've done this with black beans (they turned out to be wax beans), pinto, lima, cowpeas, kidney, and great northern beans. I've also tried seeds from other grocery items-celery seed, fennel seed, chamomile, coriander (cilantro), quinoa, fenugreek, barley, red wheat,
flax, and from scratch grains: wheat, soy beans, and sorghum. Then there is the seed and planting stock from the fresh produce: garlic, potato, sweet potato, horseradish, ginger, peppers, tomato, winter squashes, and melons. Some will show up, some won't. You can get the crops, but the name of the cultivar will be elusive. Bear in mind that plants produced from grocery store sources were produced favoring the traits of transportability and shelf appearance.
Garlic
this works just fine
Onion
when making dinner, slice off the
root section, place in a dish of
water. It will send out
roots in a few days. Keep it moist, it will send up a leaf. transplant to the garden.
Carrot
Cut off the big end where the leaves come out, place in a dish of water. When leaves reach an inch or more, transplant to garden.
I hear of this lady in Cleveland that has eaten the same carrots 5 times.
Potato
These are usually treated with sprout inhibitors. If so, growth will be slow, with dismal results. Organic potatoes do well.
Sweet Potato
Elise is the expert.
Sweet Potato Experiment.
Herbs
Lots of varieties will grow from cuttings. Several herbs are sold as cuttings rather than just leaves. Mint, thyme, rosemary, oregano, among others.