As miniature fibre-optic platforms, micro/nanofibres (MNFs) taper-drawn from silica fibres have been widely studied for applications from optical sensing, nonlinear optics to optomechanics and atom optics. While continuous-wave (CW) optical waveguiding is frequently adopted, so far almost all MNFs are operated in low-power region (e.g., <0.1 W). Here, we demonstrate high-power low-loss CW optical waveguiding in MNFs around 1550-nm wavelength. We show that a pristine MNF, even with a diameter down to 410 nm, can waveguide an optical power higher than 10 W, which is about 30 times higher than demonstrated previously. Also, we predict an optical damage threshold of 70 W. In high-power CW waveguiding MNFs, we demonstrate high-speed optomechanical driving of microparticles in air, and second harmonic generation efficiency higher than those pumped by short pulses. Our results may pave a way towards high-power MNF optics, for both scientific research and technological applications.