Elipsoid does not posess unbounded geodesics with no self-intersection. 

I do not know a conceptual explanation. 
My explanation   is that (due to integrability of the geodesic flow of ellipsoid)  we know the geodesic of the ellipsoid, let me shortly describe them. 

 The  typical geodesics viewed as a curve in the tangent bunlde lives on the Liouville torus and is a winding -- periodic or quasiperiodic -- on it. The projection of the Liouville torus to the ellipsoid is a ring (the projection is singular at two lines which project to the boundary circles of the ring and 
 otherwise is the double cover of the interior of the ring. This implies that each such typical 
 geodesic  intersects itself.


Consider now ``untypical geodesics'', i.e., those such that their lift to the tangent bundle lies on a singular leaf of the liouville foliation or is a critical circle.  The second type  are already  closed geodesics ( and on the ellipsoid there are at  most  3 of  such geodesics of the second type). 

 Now,  the last case, i.e. the  geodesic lying on a critical leaf and precisely the geodesic passing through 4  umbillic points,
 and we know that if a geodesic passes  an umbilic point of the ellipsoid it passes through infinitely   umbilic points infinitely many times which implies it has selfintersections.