This is something I want to try this year. Apparently, you can grow up to 100 pounds of potatoes in just 4 square feet of land! How? By growing them vertically in a box.
Basically, you build a box with removable panels. As the potato grows, you add panels, and then fill it in with dirt. If you buy a late-season potato, it should produce potatoes all through the box as it grows.
Details are here (also here). And someone had an extended blog entry about their own experiences here.
In short,