Since the average of $m$ and $n$ is 5, then $\frac{m+n}{2}=5$ which means that $m+n=10$. In order for $n$ to be as large as possible, we need to make $m$ as small as possible. Since $m$ and $n$ are positive integers, then the smallest possible value of $m$ is 1, which means that the largest possible value of $n$ is $n=10-m=10-1=9$.