Note that $n^{200}=(n^{2})^{100}$ and $3^{500}=(3^{5})^{100}$. Since $n$ is a positive integer, then $n^{200}<3^{500}$ is equivalent to $n^{2}<3^{5}=243$. Note that $15^{2}=225,16^{2}=256$ and if $n \geq 16$, then $n^{2} \geq 256$. Therefore, the largest possible value of $n$ is 15.