Let $S$ be a surface embedded in $\mathbb{R}^3$. A simple geodesic on $S$ is one that does not self-intersect. Some surfaces have simple geodesics whose length exceeds any given bound $L$. For example, a cylinder or a torus allows tight winding geodesics that are arbitrarily long before they cross themselves. But a sphere, or a Zoll surface, does not admit arbitrarily long simple geodesics, because every geodesic forms a simple closed loop.

Q. Which surfaces $S$ admit arbitrarily long simple geodesics?

To be specific: Do ellipsoids possess such geodesics?