More precisely, does there exist a compact surface-with-boundary $\Sigma$ with the following property? For every $g\geq 0$, there exists a subsurface-with-boundary $\Sigma'\subseteq\Sigma$, where the genus $g'$ of $\Sigma'$ is bigger than $g$?
My intuition tells me that it cannot, but I can't find a proof of this fact. Is it true, and if so, how do I prove it?
This cannot happen, because the genus $g$ of a compact oriented surface-with-boundary $\Sigma$ is equal to the largest cardinality of a set $c_1,\ldots,c_g \subset \text{interior}(\Sigma)$ of simple closed curves which are pairwise disjoint and nonseparating, meaning that $\Sigma - (c_1 \cup\cdots\cup c_g)$ is connected. So if $\Sigma' \subset \Sigma$, and if $g'$ is the genus of $\Sigma'$, and if $c'_1,\ldots,c'_{g'}$ are pairwise disjoint and nonseparating in $\Sigma'$, then they are also pairwise disjoint and nonseparating in $\Sigma$, hence $g' \le g$.