Linked from
The 9 pages that link to Rydberg constant, each with the reason it gives.
Lyman seriesRelated: It fixes the wavelength scale in the Rydberg formula for this series.
Electron massRelated: Its theoretical value depends on electron mass and reduced mass.
Theodor W. HänschRelated: High-precision hydrogen spectroscopy constrains its value.
Exotic atomRelated: Exotic-atom spectra provide independent constraints on this constant.