I've heard the statement:
Let $S$ be a hyperbolic surface with boundary and $\left[ \alpha \right] \in \pi_1 \left( S \right)$ a non-trivial element that is not conjecture to boundary element, then there is some $ f \in Mod(S) $ such that the orbit of $\left[ \alpha \right]$ under $f_*$ intersects infinitely many conjugacy classes, i.e. the set
$$ \left\{ \left[ \left( f_* \right) ^{(n)} \left( \alpha \right) \right]_{conj} \mid n\in \mathbb{N} \right\} $$
is infinite.
Slight changes to the condition of the statement might be needed (I'm positive the statement is true, but state me corrected if its not).
There is a slightly nicer formulation of this statement with homotopy without base point, but I prefer this form.
I could not find any reference for this, and could not prove this myself, does anyone know how to prove this or can refer me to some proof?