We report on efficient generation of millijoule-level, kilohertz-repetition-rate few-cycle laser pulses with radial polarization by combining a gas-filled hollow-waveguide compression technique with a suitable polarization mode converter. Peak power levels >85 GW are routinely achieved, capable of reaching relativistic intensities >10(19) W/cm2 with carrier-envelope-phase control, by employing readily accessible ultrafast high-energy laser technology.