Years ago I've grew potatoes from the peels after paring potatoes for dinner. I used to add them to a small compost pile. Next thing I knew I had potato vines. I harvested them as small "new" potatoes and got about two pound.
Normally I would buy a pound of seed potatoes but I've also planted those from the bin that had sprouted. I prefer to grow my potatoes under a mulch layer of straw, leaf and grass clippings rather than in hills. It takes a bit more space but it's easy to lift the edge of the mulch to gather a few small potatoes from time to time. Since the reason I grew potatoes was to have those small potatoes it was the best solution for me. Storage potatoes are plentiful and relatively cheap wher I live so I didn't need a large harvest of keepers.