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 6, the largest possible degree of $q(x)$ that will allow the function to have a horizontal asymptote is $\boxed{6}.$

To see that a degree of $6$ works, we can consider $q(x)=x^6.$ Then as $x$ grows away from $0,$ the $x^6$ terms in the function will dominate and the function will tend towards $\frac{x^6}{x^6}=1.$