High-power, octave-spanning supercontinuum generation through pumping a highly nonlinear fiber with noise-like and well-defined optical pulses is investigated. A two-stage fiber amplifier, consisting of a pre-amplifier and a booster, is used to greatly enhance the average power of noise-like pump pulses from 14 mW to 13.1 W. Owing to the limited coupling efficiency, only a maximum optical power of 4.63 W is launched into the nonlinear fiber. As a result, a supercontinuum spectrum spanning from 940 to 2300 nm with an average power of 3.82 W is achieved with noise-like pump pulses. This is, to the best of our knowledge, the highest average power obtained over such an octave-spanning supercontinuum spectrum using noise-like pump pulses. Supercontinuum generation with a broader spectral range and a higher average power is feasible if the coupling efficiency of the pump pulse power launched into the nonlinear fiber is increased. For the purpose of comparison, well-defined pump pulses with a similar average power at a similar repetition rate are also investigated for similar octave-spanning supercontinuum generation. A supercontinuum spectrum spanning from 950 to 2500 nm with an average power of 3.62 W is obtained with well-defined pump pulses. Even though the spectrum of the resulting supercontinuum using the well-defined pump pulses is broader, a large portion of its optical power is concentrated around the center wavelength, leading to a lower spectral power density at other wavelengths; furthermore, a higher extent of spectral structure appears in its optical spectrum, resulting in a higher power variation among different wavelengths.