Imagine that a certain animal's Fur (S) length is determined by three genes. The genotype for long fur is AABBCC, short fur is aabbcc. A cross between long fur and short fur produces f1 offspring with medium-length fur (AaBbCc). What is the Punnett square that shows a cross between two F1 off spring. How many phenotypes are possible?

As you pointed out, the F1 genotype is AaBbCc. Each F1 can produce eight unique gametes, which are equally probable: ABC, ABc, AbC, aBC, abC, aBc, Abc, and abc. Thus a cross between two F1 individuals leads to an 8x8 Punnett Square. I've included an image. I put in the same set of possible gametes for each of the two F1 parents.
Inferring from the question, each of the three loci contributes equally to the fur length. If this is correct, the phenotypes may be identified by the number of dominant genes in each. Offspring can have from zero (aabbcc) to six (AABBCC) dominant alleles. You didn't ask for the relative numbers of each phenotype, only the number possible. All combinations of A, a, B, b, C, and c are possible, so every number (of dominant alleles) from zero to six can occur. That is seven different phenotypes. If you wanted to describe them in words, you could call them:
long (AABBCC; six dominant alleles)
almost long (AaBBCC, AABbCC, AABBCc; five dominant alleles)
medium-long (four dominant alleles)
medium (same as F1, plus other combinations with 3; see Punnett square)
medium-short (two dominant alleles)
almost short (one dominant allele)
short (aabbcc)