Motivated by the possibility of guiding daughter ions from double beta decay events to single-ion sensors for barium tagging, the NEXT collaboration is developing a program of R&D to test radio frequency (RF) carpets for ion transport in high pressure xenon gas. This would require carpet functionality in regimes at higher pressures than have been previously reported, implying correspondingly larger electrode voltages than in existing systems. This mode of operation appears plausible for contemporary RF-carpet geometries due to the higher predicted breakdown strength of high pressure xenon relative to low pressure helium, the working medium in most existing RF carpet devices. In this paper we present the first measurements of the high voltage dielectric strength of xenon gas at high pressure and at the relevant RF frequencies for ion transport (in the 10 MHz range), as well as new DC and RF measurements of the dielectric strengths of high pressure argon and helium gases at small gap sizes. We find breakdown voltages that are compatible with stable RF carpet operation given the gas, pressure, voltage, materials and geometry of interest.