Limit on the Number of Facial Phenotype
Okay, theoretically possible, but only in theory.  Practically, it's totally impossible.  Let's start by defining why it would theoretically be possible.
Theoretically, there are only so many possible arrangements of proteins in the face, which would represent a limited number of possible facial appearances.  However, as you probably guessed, there are zillions of proteins being expressed in each body organ, such as an eye.  So just from the perspective of the number of different possibilities, it would be impossible.
One might think that since the facial features are controlled by genetic variation, eventually these features will begin to repeat themselves, but such is not the case, for two reasons:

There are way to many genes - and alleles - that control facial features.  We're talking thousands if not millions.  When you raise that number of alleles to each others' powers, you start to see the impossibility of repeating the exact same combination, especially since every new generation is introducing new alleles from a new blood line.

Genetic code is not the only thing that changes how you look.  A whole branch of genetics, epigenetics, is devoted to the changes of gene expression unrelated to DNA sequence.  The expression can be changed by environmental factors, even as early as when the embryo is forming the placenta.  This is why identical twins (who have identical DNA) do not actually look exactly the same, or even have identical fingerprints.



Bottom line, totally impossible.

Sources:

Face shape is in the genes, PLOS, 2016
Epigenetics, Wikapedia, 2017
Why do identical twins have physical differences?, Anonymous, 2005
Introduction to epigenetics, University of Leicester, 2013
How many Proteins exist in human body?, Innovateus, 2013