Lunes http://cs.smith.edu/%7Eorourke/MathOverflow/Lunes.jpg
I believe this is what vlsd must mean by "the intersection of the three lunes."
While I'm posting this clarifying (I hope!) image, I'll risk a heuristic argument
to support
why I think it is "possible to cover the entire sphere using this method":
(1) A region on the sphere is only a fixed point with respect to reflection in the three planes if the three great circles meet in one point. For example, a geodesic equilateral triangle is symmetric with respect to three circles meeting at its centroid, and would be fixed w.r.t. reflection in those three planes. In other words, a shape can only be fixed w.r.t. to three lines of reflection if those lines meet in a point. I assume that vlsd would exclude this degenerate situation (else the three-lune intersection is a point).
(2) Since the region is not a fixed point w.r.t. reflections (if (1) is correct), the only way the process could avoid covering the sphere is if repeated reflections and unions approached a limit shape. But the displacement of points in the reflecting region is lower-bounded by some function of the smallest angle between the planes, so I do not believe this can occur.
I am aware this is not a precise argument!