Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$ that has a geodesic $\gamma$ that spirals around a point $x$, getting closer and closer, but never reaching $x$?

Here I am thinking of the metric on $S$ as inherited from $\mathbb{R}^3$. A portion of such a $\gamma$ might look something like this on the surface, with $\gamma$ confined to a smaller and smaller area on $S$:

Perhaps $S$ is not smooth at $x$.

**by Robert Bryant in the comments: A surface of revolution with a "pit" whose bottom is $x$ could support a geodesic that spirals down the pit. "If the geodesic actually limits to $x$, ...[then $x$ cannot be] a smooth point of the surface... However, if the geodesic is instead limiting to a (very small) closed geodesic encircling $x$, ... then $x$ could be a smooth point."**

*Answered*