We present evidence of all-optical trion generation and emission in pristine single-walled carbon nanotubes (SWCNTs). Luminescence spectra, recorded on individual SWCNTs over a large cw excitation intensity range, show trion emission peaks redshifted with respect to the bright exciton peak. Clear chirality dependence is observed for 22 separate SWCNT species, allowing for determination of electron-hole exchange interaction and trion binding energy contributions. Luminescence data together with ultrafast pump-probe experiments on chirality-sorted bulk samples suggest that exciton-exciton annihilation processes generate dissociated carriers that allow for trion creation upon a subsequent photon absorption event.