9. The common ratio of the geometric sequence $a+\log _{2} 3, a+\log _{4} 3, a+\log _{3} 3$ is
9. $\frac{1}{3}$

Let the common ratio be $q$. From the given conditions, we have
$$
q=\frac{a+\log _{4} 3}{a+\log _{2} 3}=\frac{a+\log _{8} 3}{a+\log _{4} 3} .
$$

According to the properties of proportions, we get
$$
\begin{aligned}
q & =\frac{a+\log _{4} 3-\left(a+\log _{8} 3\right)}{a+\log _{2} 3-\left(a+\log _{4} 3\right)}=\frac{\log _{4} 3-\log _{8} 3}{\log _{2} 3-\log _{4} 3} \\
& =\frac{\frac{1}{2} \log _{2} 3-\frac{1}{3} \log _{2} 3}{\log _{2} 3-\frac{1}{2} \log _{2} 3}=\frac{1}{3} .
\end{aligned}
$$