I found in a paper that the maximum number of orthants that an affine space could intersect is given by
$\sum_{i = 0}^d \binom{m}{i}$
with $d$ the dimension of the affine space embedded in $\mathbb{R}^m$. This seems like a result that should be easliy google-able but I did not find a proof.
Additionally, given an affine map $x \mapsto Wx + b$, how could I calculate the number of orthants occupied by the image of that map?