How many Miller Indices are there? Are there an infinite combination of (h,k,l) Miller Indices, or would there be some sort of limit to the possible combinations?
 A: Let us start from the definition of Miller indices:
the set of all the Miller indices is in one-to-one correspondence with the set of all the possible planes passing through the points of Bravais Lattice (BL) points. The BL is a theoretical concept useful for describing the translational symmetry of a crystal. A BL is infinite, contains infinite points and infinite sets of parallel planes passing through them. Therefore, there is no upper limit to the set of the possible Miller indices.
This fact can be seen easily by noting that each triple ($hkl$) denotes the family of planes orthogonal to the direction $h {\bf b}_1+k {\bf b}_2+l {\bf b}_3$, where is a basis of the Reciprocal Lattice. Linearly independent triples denote different families. Being $(hkl) \in \mathbb Z^3$, there are infinite independent triples, therefore, infinite independent directions in the reciprocal lattice and finally infinite Miller indices. There is no contradiction with the existence of a minimum distance between the sites of the BL since the sites in two nearest neighbor planes have not to be one in front of the other.
In conclusion, it is not possible to list all the Miller indices. What can be done is listing those corresponding to all the planes with an interplanar distance larger than some threshold.
A final remark on finite crystals. Of course, every real crystal has to be finite, and this trivial fact would induce a natural cut-off of the values of Miller indices. However, for macroscopic crystals, a direct connection between the crystal size and the upper limit of the Miller indices is not usually done.
