It depends on where you live. Asparagus is a hardy perennial. So is rhubarb, although some people don't consider it a vegetable. Peppers and tomatoes are tender perennials, so if you live in a frostfree climate, they will continue to grow as long as they have enough water.
Many tubers will grow indefinitely without replanting, as long as some are left in the ground, like potatoes or Jerusalem artichokes.
Some plants are biennials, and they will grow 2 years without replanting, like carrots, parsnips, beets, cabbages, broccoli, kale, parsley, onions, leeks, garlic.
Other plants are self-seeders, and will continue to produce on their own without planting, such as tomatoes, amaranth, pumpkins, dill, coriander, tomatillos, and husk cherries.
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