From TipNut

Quite the clever gardening tip here folks! Today’s feature includes tips from three different sources for growing potatoes vertically (in layers) instead of spread out in rows across your garden. If you have limited garden space or want to try some nifty gardening magic, this could be a great option for you.

Seattle TimesFirst, there’s this article from The Seattle Times: It’s Not Idaho, But You Still Can Grow Potatoes:

The potatoes are planted inside the box, the first row of boards is installed and the dirt or mulch can now be added to cover the seed potatoes. As the plant grows, more boards and dirt will be added.

You plant in one bottom layer, boarding up the sides of each layer and adding dirt as you go higher (you wait until the plants have grown a bit before adding a new layer). While new potatoes are growing in the top layers, remove the boards from the first layer at the bottom to carefully dig out any that are ready for harvesting. Fill the dirt back in and board up the box again. You move up the layers and harvest as they are ready. I imagine the new potatoes in the first couple bottom layers would be somewhat awkward to get at but as you move higher–not so bad…. read more here