Suppose we tile the plane with a simply-connected tile $T$, and we wish to ensure that every tile touches at least $k$ other tiles.
If we restrict "touching" to require sharing a positive-measure section of their border, then Euler's formula for planar graphs proves that we can only get $k=6$, a value which is attained by taking $T$ to be the regular hexagon.
However, if we permit adjacency by sharing a corner, we can do better. For instance, the standard tessellation of the plane by squares has $k=8$, and the standard tessellation by equilateral triangles has $k=12$. In fact, one can find a tiling with $k=21$, shown below:
All of the above examples are isohedral, which means that any two tiles can be mapped to each other via an isometry of the plane preserving the tiling. We can consult The eighty-one types of isohedral tilings in the plane by Grünbaum and Shephard to confirm that no other isohedral tilings improve on this value.
Is $\textbf{k=21}$ maximal among monohedral tilings? (A monohedral tiling is one in which we merely require all tiles to be congruent.)
Note that it is possible to find monohedral tilings in which some tiles have arbitrarily high corner-adjacency (or edge-adjacency, for that matter), but all such tilings I have found contain other tiles which are not so well-connected.
I'm interested in any improvements to the above $k=21$ solution, and any upper bounds that can be put on $k$ in the monohedral case.