I've done the potato barrel experiment for two summers now.
Year one was when I just dropped a potato with three eyes in one of my planters and walked away. I wasn't there to care for it or water it because my house was undergoing major renovations. The eyes grew out for two or three feet before the whole thing died. When I threw it out I noticed something odd in the dirt. It turned out to be one of three potatoes about the size of a beer bottle cap.
I held on to two of them. Instead of rotting they sat happily on the shelf all winter long. One started to sprout eyes this spring. I nurtured it along for a couple of weeks and then planted it in my potato barrel.
The one eye grew up for about a foot and then fell over. I put some dirt on it and waited. It grew up about a foot and fell over. I put more dirt on it. It grew up about a foot and fell over. I put more dirt on it. I was hoping that part of the potato vine would split off in another branch. It never happened. I was also going for a corkscrew effect. I figured I'd get a barrel full of dirt and the potato vine rising up a loop at a time.
Instead, about the fifth time I put dirt on it, the plant died.
When knocked over the barrel I found that I HAD successfully converted some of the worst soil I've ever seen outside of a clearcut rain forest into the richest, blackest, moistest soil I've ever seen full of the biggest earthworms I've heard of in the United States. That's because I threw in a mixture of a mostly sand type of dirt collected from the alley, grass clippings, apple cores and orange peels, a few worms, and probably a handful of goat chow or two.
I wound up dumping a few handfuls of this stuff on all my other plants.
So here's the strategy for next summer:
1) I wasn't supposed to use just one potato to start. I could have, but I treated it wrong and failed. No, I should have either had one potato with several eyes and cut them apart into separate seed plants OR had several potatoes each with eyes. I think six total eyes would be right.
2) Don't allow the plants to fall over. As they grow up and up and up keep feeding dirt, vegetables, compost, etc. so they always have support. On a potato farm they'll want to send runners across the soil. But in a barrel they don't have that luxury so keep them going up.