We have investigated the photophysical properties of heavily doped semiconducting single-wall carbon nanotubes (s-SWNTs) in the vis and NIR spectral range. Doping is here achieved either by electrochemical gating or by redox chemistry, using AuCl₃ as oxidizing agent. Stationary and time-resolved absorption spectroscopy show that the dynamical behaviour of electrochemically- and redox-doped s-SWNT samples are identical, indicating that doping mechanisms and the character of charged states is the same for both doping processes. Spectroscopically, the heavily doped regime is defined by the disappearance of NIR trion and neutral exciton signatures. The remaining absorption band in heavily hole-doped (6,5) s-SWNTs is broad, flat and practically featureless over a very broad spectral range from about 1150 nm to 620 nm. It carries nearly all the oscillator strength of the first subband exciton of intrinsic s-SWNTs. The transient spectral response of this so-called H-band (for heavily doped) is surprisingly weak, suggesting that signal contributions from ground- and excited state absorption and from stimulated emission cancel out nearly perfectly. We discuss these findings also in the context of UV-IR measurements of the optical conductivity which shows an ambiguous response in the low THz regime.