  The nature of the theory of circular Rydberg states of hydrogenlike ions
allows highly-accurate predictions to be made for energy levels. In particular,
uncertainties arising from the problematic nuclear size correction which beset
low angular-momentum states are negligibly small for the high angular-momentum
states. The largest remaining source of uncertainty can be addressed with the
help of quantum electrodynamics (QED) calculations, including a new
nonperturbative result reported here. More stringent tests of theory and an
improved determination of the Rydberg constant may be possible if predictions
can be compared with precision frequency measurements in this regime. The
diversity of information can be increased by utilizing a variety of
combinations of ions and Ryberg states to determine fundamental constants and
test theory.
