jmc

For the given function to have a horizontal asymptote, it can't go to infinity as $x$ goes to infinity. This is only possible if the numerator has the same or smaller degree than the denominator. Since the denominator has degree 5, the largest possible degree of $q(x)$ that will allow the function to have a horizontal asymptote is $\boxed{5}.$

We note that 5 is in fact possible, because if we take $q(x)=x^5,$ then the rational function has horizontal asymptote $y=\frac{1}{2}.$