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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 $\tfrac1r$.