I've heard that Red Pontiac is a good one.

"Late-Season Potatoes"
Elba Lehigh Bintje
Canela Russet Desiree German Butterball
Green Mountain Katahdin Red Maria (a.k.a NY 129)
Red Pontiac

You need to check on the reds. As far as I ever heard or experienced, they will only produce in the first 6". I just grow white Kennebec in my potato bin. They will continue growing all the way up as long as you keep filling the bin with soil up to the top couple of leaves whenever the plants reach arount 7" tall. I will grow until just before a hard frost.

The only potato I know for a fact will produce beyond the bottom level in a tower is a Russet Burbank, given the right conditions. Most potatoes are essentially determinate, producing all their spuds in one flush. Given enough time (probably at least 135-150 days) russets can produce several flushes.