Question

In addition to Mosher's examples:

Take the surface of tubular neighborhood $\varepsilon$-neighborhood $S_\varepsilon$ of a smooth curve $\gamma$(*) for some $\varepsilon < r$. It will have exactly one closed geodesic through any point and the rest will be simple infinitely long geodesics.

Indeed, any geodesic is either a meridian of $S_\varepsilon$ or it intersects transversely any meridian and therefore it has no self-intersections.

$(*)$ I assume that $\gamma$ is an infinite curve such that for any point $p$ on distance $ < r $ from $\gamma$ there is unique point $\bar p\in\gamma$ which minimize the distance $|p-\bar p|$. In particular the curvature of $\gamma$ has to be less $\tfrac1r$.