Here are some variations on go 囲碁 boards. Go can be played on arbitrary graphs, but we mostly stick with geometries in which each point (node) has 4 neighbors.

Traditional 19x19 square go board but with 1 corner point removed. May need to significantly adjust komi because the first player can grab the 2 "easy" corners.

Start with a square but remove a small square or rectangle of points from a corner, so L shaped board. This introduces 2 more corners and 1 concave corner (270 degree vertex).

Two square boards sharing (overlapping at) a corner point.

Two square boards overlapping in a square or rectangle of points around a corner.

Two square boards overlapping at all 4 corner points. This requires drawing on a spherical manifold.

Two square boards overlapping at all corners and edges. Again requires a spherical manifold.

We next consider several square boards joined as in the faces of a cube:

3 squares around a vertex of a cube. This introduces one point (at the vertex) with 3 neighbors.

4 squares: 2 adjacent to an edge and another 2 whose corner is an endpoint of the edge.

5 and 6 squares (previously discussed). Some creativity possible in drawing these manifolds on a flat surface or display.

We can also consider a diamond-shaped board. The diamond is equivalent to considering only the squares of one color on a checkerboard. Edge play will be very different; there are corners along every edge.

Repeat all the above modifications to square boards with diamond boards (turned 45 degrees so that it is square again).