I used square wire cages without a top or bottom and filled with dry leaves.
Excellent information. Thanks for sharing your research.Just thought I would share my 2 cents worth of research on the topic. In short, the best potatoes to use for the "grow-up method" would be "late" potatoes. Something like German ButterBall, Red Pontiac, Kennebec, etc. The reason in simple terms - early and mid season taters set all the potatoes in a short period of time. Sort of like a determinate tomato. Late season taters continually develop potatoes similar to an indeterminate tomato.
If you plant a late season potato, as you cover the foliage, it keeps grwoing, therefore, it should continue to set potatoes all the way until it dies.
Early and Mid season couldn't care less. When there done, there done. The foliage could be 10 foot tall, but all the potatoes will be at the bottom.
Your mileage may vary, but typically this is why one person has good luck, and another doesn't. This I plan to try both early and late potatoes to test this out.