We demonstrated an optimization of a picosecond fiber amplifier based on Yb-doped tapered fiber in a spectral range of 1030 nm. Nonlinear effects limiting peak power scaling (stimulated Raman scattering and four-wave mixing) were studied and factors affecting their threshold were established, such as gain, diameter profile along the length of taper, output mode field diameter, and numerical aperture of a pump. By determining the optimal amplification regime and manufacturing advanced tapered fibers, we amplified 13 ps pulses to a record-high peak power of 1 MW at a wavelength of 1029 nm directly at the output of the fiber at an average power of 13.8 W. Four-wave mixing was the limiting factor, and the total fraction of deleterious components in the output spectrum was ~2%. The quality of the output beam was close to being diffraction limited (M2 < 1.2).