Surfaces filled densely by a geodesic
Any surface of revolution in $3$-space with poles will have this property. The reason is that, in this case, any geodesic either goes through a pole (i.e., a point where the axis of revolution meets the surface) and is a profile curve that lies in a plane or else, because of the Clairaut integral, it avoids that pole by some positive distance. Thus, no geodesic on the surface is dense in the surface.
You mention ellipsoids, which furnish examples of these special surfaces. These are examples of so-called 'Liouville surfaces', i.e., Riemannian surfaces $(S,g)$ for which there exist two independent first integrals of the geodesic flow on $T^\ast S$ that are quadratic functions on the fibers of $T^\ast S\to S$, one of which is the co-metric associated to $g$ and the other of which is an independent first integral. As you probably know, surfaces of revolution are surfaces for which there exist a first integral of the geodesic flow that is linear on the fibers of $T^\ast S\to S$, namely the Clairaut integral. It has been known for some time that there are metrics on the $2$-sphere that don't possess any 'extra' first integrals that are linear or quadratic functions on the fibers of $T^\ast S\to S$, but do possess first integrals that are cubic or quartic functions on the fibers of $T^\ast S\to S$. These are due to Goryachev-Chaplygin (early 20th century) and Dullin-Matveev (2004). These are also examples for which no geodesic winds densely over the surface. All of these work because there are 'conservation laws' for the geodesic flow of a particular kind, and they properly generalize the Liouville surfaces (which includes the famous case of ellipsoids).
Donnay and Pugh proved that every embedded surface $S\subset R^3$ can be $C^0$-perturbed so that the new metric has ergodic geodesic flow, see here. In particular, the new metric will have dense geodesics (moreover, "generic" geodesics will be dense in the unit tangent bundle).
Clairaut's relation shows that any simply connected surface of revolution has this property, whether convex or not (for the reason stated by Robert).