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A novel, all-fiber-integrated supercontinuum (SC) laser is demonstrated and provides up to 10.5 W time-averaged power with a continuous spectrum from ~0.8 to 4 mum. The SC is generated in a combination of standard single-mode fibers and ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAN) fluoride fibers pumped by a laser-diode-based cladding-pumped fiber amplifier system. The output SC pulse pattern can be modulated by directly modulating the seed laser diode. Near-diffraction-limited beam qualities are maintained over the entire SC spectrum. The SC average power is also linearly scalable by varying the input pump power and pulse repetition rate. We further investigate the theoretical limitations on the achievable average power handling and spectral width for the SC generation in ZBLAN fibers. Based on the thermal modeling, the standard ZBLAN fiber can handle a time-averaged power up to ~15 W, which can be further scaled up to ~40 W with a proper thermal coating applied onto the ZBLAN fiber. The SC long-wavelength edge is limited by the nonlinear wavelength generation processes, fiber bend-induced loss, and glass material loss. By using a ZBLAN fiber with a 0.3 numerical aperture, the SC spectrum could extend out to ~4.5 mum, which is then limited by the material loss.