Step 1: Grow UP!
Step 2: Grow up with a potato tower
Make a tube of fencing (as seen below), or out of a trash can with the bottom cut off. Plant your potatoes as you would normally, but more densely (5-10, depending on diameter) in an area the size of your tube. As your potatoes grow, add alternating layers of straw and compost, leaving about 6" of leaves exposed at all times. At the end of the growing season, when the plant leaves start to die back, you can kick over the tower and pull out your harvest of potatoes.