Let $a$ and $b$ be two relatively prime natural numbers. What is the largest number $c$ such that there is a curve with maps to $\mathbb P^1$ of degree $a$ and $b$ but no map to $\mathbb P^1$ of degree $c$?
It's easy to seee that a map of degree $a$ and $b$ implies a map of degree each nonnegative integer combination of $a$ and $b$. So $ c \leq ab-a-b$, the largest number that is not a nonegative integer combination of $a$ and $b$.
But I don't think that's optimal. Given a map of degree $a$ and a map of degree $b$, we get a map of degree $(a,b)$ to $\mathbb P^1 \times \mathbb P^1$ which has a Segre embedding into $\mathbb P^3$. By projecting from a line which intersects the curve at $0,1,$ or $2$ points, we can get maps of degree $a+b$, $a+b-1$, and $a+b-2$. By iterating this we can always get a much smaller upper bound. So what's the optimal bound?