Let $\Sigma$ be a compact orientable connected $2$-manifold with a non-empty boundary. Let $\widehat \pi(\Sigma)$ denote the set of free homotopy classes of
curves in $\Sigma$. We say $x\in \widehat \pi(\Sigma)$ is peripheral, if there is representative $\alpha\colon \Bbb S^1\to \Sigma$ of $x$ with $\text{im}(\alpha)\subseteq \partial \Sigma$.



> 
> **Question:** Let $x\in\widehat \pi(\Sigma)$ be a *non-trivial non-peripheral* element. Does there exist a simple closed curve
> $\beta\subseteq \Sigma$ such that the geometric intersection of
> $\alpha$ and $\beta$ is non-zero, where $\alpha$ is a representative
> of $x$.
> 
> 
> 



Characterization of a non-trivial non-peripheral element of the free homotopy classes of a compact bordered surface
This is true for all (compact, connected, oriented) surfaces that admit essential non-peripheral simple closed curves.


The sphere, disk, annulus and pants do not admit essential non-peripheral curves. The sphere and disk have trivial fundamental group. So the answer is "yes" vacuously for the sphere and disk, and "no" for the annulus and pants (as suggested by Johannes’).


Suppose that $S$ is any other compact, connected, oriented surface.
Now $S$ admits *filling laminations*. Any sufficiently good (Hausdorff close) simple closed curve approximation to a filling lamination does what you want. The existence of such laminations is a consequence of Thurston’s theory of pseudo-Anosov maps.


