$R^3$ can be tiled regularly with cubes, where each tile has six neighbors. If I do not care for the form of the tiles, but only for the number of neighbors, is there a limit on the number of neighbors if each tile shall have the same number of neighbors?
Since the form of the tiles is not important, this may well be eqivalent to the question about the degrees in an infinite non-planar graph where all degrees shall be the same.
Use $n\times 1\times1$ blocks ("prismatic sticks"), forming horizontal layers of thickness $1$. The sticks in layers $2k-1\leq z\leq 2k$, $\>k\in{\mathbb Z}$, are parallel to the $x$-axis, and the sticks in layers $2k\leq z\leq 2k+1$ are parallel to the $y$-axis.