At the Tiling Database there are 3, 20, 198, 1390 (A054361) non-tiling polyominoes of order 7 to 10.
In 3D, solids with a particular Heesch number don't seem to be well known. Glenn Rhoads found the minimal Heesch 2 polyomino in his 2003 thesis "Planar Tilings and the Search for an Aperiodic Prototile".
If this polyform is modified to be a polycube, can it fill space? If not, the 3D Heesch number is at least 2 and possibly 3. That basically uses layering of an existing 2D Heesch object. What is the Heesch number for this polycube?
If thickened 2D Heesch objects are excluded, what are the simplest polyhedra, polyforms, and shapes which have 3D Heesch numbers 1, 2, 3?
For a 3x3x3 cube, add an outie 1x1x1 cube on one face center and remove innie 1x1x1 cubes from centers of other faces. Stack 27 of these with the outie up. Rotate four of them to fill the center innies. The center shape is completely surrounded, showing a purely 3D Heesch-1 object.
Here is a Heesch-2 object that likely gets weird in 3D.
