jmc

The formula for the interior angle of a regular sided polygon is $\frac{(n-2)180}{n}$. Thus, $\frac{\frac{(r-2)180}{r}}{\frac{(s-2)180}{s}}=\frac{59}{58}$. Cross multiplying and simplifying, we get $\frac{58(r-2)}{r}=\frac{59(s-2)}{s}$. Cross multiply and combine like terms again to yield $58rs-58\cdot 2s=59rs-59\cdot 2r⟹118r-116s=rs$. Solving for $r$, we get $r=\frac{116s}{118-s}$. $r\ge 0$ and $s\ge 0$, making the numerator of the fraction positive. To make the denominator positive, $s<118$; the largest possible value of $s$ is $117$. This is achievable because the denominator is $1$, making $r$ a positive number $116\cdot 117$ and $s=\boxed{117}$.