  We address several specific aspects of the following general question: can a
field K have so many automorphisms that the action of the automorphism group on
the elements of K has relatively few orbits? We prove that any field which has
only finitely many orbits under its automorphism group is finite. We extend the
techniques of that proof to approach a broader conjecture, which asks whether
the automorphism group of one field over a subfield can have only finitely many
orbits on the complement of the subfield. Finally, we apply similar methods to
analyze the field of Mal'cev-Neumann "generalized power series" over a base
field; these form near-counterexamples to our conjecture when the base field
has characteristic zero, but often fall surprisingly far short in positive
characteristic.
