How about 11x11 variant of Go (plain, not toroidal)? 11x11 has 50% more cells than 9x9 so it should be considerably deeper while at the same time not excessively long. For me 13x13 is already quite a heavy game. 11x11 seems to strike a perfect balance of depth vs. time of play.
I would like a 7x11 board, but with the points rearranged a bit: Link
I think it would be fun to try this out. While Toroidal Go makes the board most regular, with full symmetry between the points, it would be interesting to go in the opposite direction as well and make the board more irregular.
BTW: I wonder why Richard does not tidy things up and make the board size a game parameter, as it is already in Dots&Boxes and Havannah. And why is Toroidal Go 11x11 a variant of Go19?
BTW2: at www.dragongoserver.net Go can be played on all sizes from 5x5 to 25x25. But there are no tournaments.