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We construct a class of polycubes that tile the space by translation in a lattice-periodic way  and show that for this class the number of surrounding tiles cannot be bounded. The first  construction is based on polycubes with an $L$-shape but with many distinct tilings of the  space. Nevertheless, we are able to construct a class of more complicated polycubes such that  each polycube tiles the space in a unique way and such that the number of faces is $4k+8$ where  $2k+1$ is the volume of the polycube. This shows that the number of tiles that surround the  surface of a space-filler cannot be bounded.

How many Faces can the Polycubes of Lattice Tilings by Translation of ${\mathbb R}^3$ have?