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](http://mathoverflow.net/q/28622/6094),
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?