TITLE: Surfaces filled densely by a geodesic
QUESTION [38 upvotes]: Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
  single geodesic $\gamma$ that fills $S$ densely?

Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
through which $\gamma$ passes equals $S$.
Some examples:

 A sphere: every geodesic is a great circle.
 Zoll surfaces, as discussed here:
"Surfaces all of whose geodesics are both closed and simple."
 An ellipsoid.


(Image from GeographicLib.)


 A torus generally has many geodesics that fill the surface.



(Image by John Oprea)


My assumption is that almost all surfaces have geodesics that fill them.
Is this known, under any interpretation of "almost all"?
I would also be interested in extending the list of exceptional surfaces
beyond {sphere, Zoll, ellipsoid}.
Thanks for pointers!

Answers Summary (18Apr2013):

 (Robert Bryant, Mikhail Katz)
Any surface of revolution with poles has no dense geodesic.
This holds for convex or nonconvex surfaces of revolution.
 (Robert Bryant)
There are generalizations of
Liouville surfaces (due to Goryachev-Chaplygin and to Dullin-Matveev) that have no dense geodesic.
 (Misha Kapovich)
Every surface may be perturbed by gluing on "focusing caps" so 
that it has dense geodesics.
 (Keith Burns) Guess:
There is always a dense geodesic on a closed Riemannian surface of genus 
$\ge 2$.

REPLY [4 votes]: I feel there is some implicit idea here but nobody formulated it, so let me try:
a smooth closed surface has no filling geodesic if and only if its geodesic flow is integrable
What do you think about this conjecture ?
For me it seems probable since integrable flow have their Arnold tori that don't fill up all the space: for example, for the ellipsoid, the picture of the space that a geodesic fills in is a projection of an Arnold tori (an annulus).
All the examples stated above are integrable, and Burns guess is nice because surfaces of big genus can't have an integrable flow (Kozlov theorem, if I am not wrong...)
I am really excited about this conjecture.