I planted in straw one year, just a dab of dirt where I put the spud piece. You do have to add some compost tea or fertilize a couple times. Had great results, held the moisture really well.
This year I'm growing potatoes in wire cages, made with 2x4 welded wire, about 3' x 3' x 3' tall. Since I have plenty of space, I probably won't try anything smaller.
I saw pictures of some that were grown in a black trash bag. They just rolled the bag down low to start, unrolling as they added soil along the way. They got at least 20# of potatoes from that one bag. It's waaay too hot here for me to try it. They'd be cooked in no time.