Most of us are familiar with lost veggies in the crisper that have either rotted or started growing. If you discover the latter, you're in luck! Past its prime, a potato with "eyes" begins to lose flavor and texture, and the nutritional profile changes as the tuber's sugars are directed to the growth of the new plant that's sprouting.
A new potato plant will happily grow on your counter for quite some time, but popped in a pot with water and sunshine, the growth becomes exponential. The little aging potato suddenly begins to produce so much more food than it would have if consumed pre-sprout. Sweet potato vines are prolific and thrive on my kind of neglect. Every day this potato yields a lush harvest of fresh heart-shaped greenery that is wonderful sauteed with onions and garlic. It's best to start with organic potatoes, free of pesticides and genetic modifications. Ornamental sweet potato plants can't boast that pedigree, so stick with potato plants rooted from the produce aisle.
After the potato leaf harvest, at the end of the growing season, don't forget to check the roots to see if bonus potatoes have formed.