Super fun and great science project for kids (some adults might have a bit of surprise).
Vegetables cannot grow vegetables. Or, can they??
It is always most fun to lead with questions. Stow away as many of these vegetable ends as possible: carrot, celery, green onion, lettuce, leeks.
Trim vegetables, do not cut roots off too close, do not cut tops too low, place in water (enough to provide moisture, but not so much as to rot the vegetable before it can grow). This is one of the great science projects. Start with questions, gather information, guess the answer, experiment and analyze & record your observations.
Trick vegetables (we already told you what will work): potatoes, sweet potatoes, cabbage, garlic, tomato, basil, zucchini or something else your local grocer or CSA producer might have available.
What grows in water? In soil? How short can you cut the carrot (leaving a bit of the shoulders and top) and still have growth? How smelly does broccoli get before you toss it into the compost pile? How many cuttings can you take (and eat) from a regenerated vegetable? Does the vegetable need nutrition beyond sunlight?
You can tackle many great concepts: growing food, food waste, compost, photosynthesis, scientific method, mathematics through graphs/charts, the power of inquiry and the even greater power of being wrong. Slightly rigging the experiment is a great tool, empowering kids to figure out how to move beyond obstacles. As hard as it may be, don't offer the answer. It is delightfully humbling how quickly these young, adept minds find the path to answers on their own accord.