We present a narrow linewidth continuous laser source with over 11 W output power at 780 nm, based on single-pass frequency doubling of an amplified 1560 nm fibre laser with 36% efficiency. This source offers a combination of high power, simplicity, mode quality and stability. Without any active stabilization, the linewidth is measured to be below 10 kHz. The fibre seed is tunable over 60 GHz, which allows access to the D2 transitions in 87Rb and 85Rb, providing a viable high-power source for laser cooling as well as for large-momentum-transfer beamsplitters in atom interferometry. Sources of this type will pave the way for a new generation of high flux, high duty-cycle degenerate quantum gas experiments.