Here's a question: I'm still following the hyperbolic game HyperRogue. It keeps growing and exploring new concepts.

One important concept is: which shapes can tile the hyperbolic plane if they have to be comprised from the basic hexagons and heptagons of the underlying (6,6,7) tessellation? Zeno has found one such configuration made from 40 cells. Clearly, all configurations that tile must have number of cells that's a multiple of 10 (with hexagons and heptagons in 7:3 ratio). Any such configuration with 10k cells has area k*pi and the resulting tiling can be considered as a variant of {3k+6,3} tessellation as the underlying (6,6,7) only permits 3-valent vertices.

So: any configuration with 10 cells (smallest possible) will have 9 neighbours around every tile (assuming uniformity). I've found a 10-cell configuration that tiles (hexagon, 3 surrounding hexagons + 3 surrounding heptagons, 3 hexagons at distance 2, looks like a fat triangle), and yesterday I discovered that it has 4 distinct ways to cover the hyperbolic plane uniformly as long as its 3-fold symmetry stays unbroken. There might be more ways to use it if you allow it to be surrounded in an asymmetric way.