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The Heesch number of a closed plane figure is the maximum number of times that figure can be completely surrounded by copies of itself. The determination of the maximum possible (finite) Heesch number is known as Heesch's Problem. The Heesch number of a Triangle, Quadrilateral, regular Hexagon, or any other shape that can Tile or Tessellate the plane, is infinity. Conversely, any shape with infinite Heesch number must tile the plane (Eppstein). The largest known (finite) Heesch number is 3, and corresponds to a tile invented by R. Ammann (Senechal 1995).
References
Eppstein, D. ``Heesch's Problem.''
http://www.ics.uci.edu/~eppstein/junkyard/heesch/.
Fontaine, A. ``An Infinite Number of Plane Figures with Heesch Number Two.'' J. Comb. Th. A
57, 151-156, 1991.
Senechal, M. Quasicrystals and Geometry. New York: Cambridge University Press, 1995.