Round Board
There is a variant of go played on a round board. Apparently it was introduced at a European Go Congress a few years ago: (dead link)
http://www.escape.de/users/shusaku/euro96/issue3/roundgo.htm
The board is an interesting spiral. A printable board can be found in PDF here:
http://www.youdzone.com/go.html
Some more mathematical information can be found here: (dead link)
http://www.leipzig-go.de/gitter/index_e.html
As of summer 2023, this game can be played on the GoVariants server, under the name badukWithAbstractBoard / circular: https://www.govariants.com/