Single, double, and triple photoionization of Ne+ ions by single photons have been investigated at the synchrotron radiation source PETRA III in Hamburg, Germany. Absolute cross-sections were measured by employing the photon-ion merged-beams technique. Photon energies were between about 840 and 930 eV, covering the range from the lowest-energy resonances associated with the excitation of one single K-shell electron up to double excitations involving one K- and one L-shell electron, well beyond the K-shell ionization threshold. Also, photoionization of neutral Ne was investigated just below the K edge. The chosen photon energy bandwidths were between 32 and 500 meV, facilitating the determination of natural line widths. The uncertainty of the energy scale is estimated to be 0.2 eV. For comparison with existing theoretical calculations, astrophysically relevant photoabsorption cross-sections were inferred by summing the measured partial ionization channels. Discussion of the observed resonances in the different final ionization channels reveals the presence of complex Auger-decay mechanisms. The ejection of three electrons from the lowest K-shell-excited Ne+() level, for example, requires cooperative interaction of at least four electrons.