The atom-atom-anion three-body recombination (TBR) of $^{3}\mathrm{He}+^{3}\mathrm{He}+{X}^{\ensuremath{-}}$($X=\mathrm{H}\phantom{\rule{4pt}{0ex}}\text{or}\phantom{\rule{4pt}{0ex}}\mathrm{D}$) systems at ultracold temperatures ($T=0.01\ensuremath{\sim}100$ mK) are studied by solving the Schr\"odinger equation in the adiabatic hyperspherical representation. It is found that for each system, $^{3}\mathrm{He}+^{3}\mathrm{He}+{\mathrm{H}}^{\ensuremath{-}}$ or $^{3}\mathrm{He}+^{3}\mathrm{He}+{\mathrm{D}}^{\ensuremath{-}}$, the ${J}^{\mathrm{\ensuremath{\Pi}}}={1}^{\ensuremath{-}}$ symmetry dominates the TBR process, and the rates of TBR into $l=1\phantom{\rule{4pt}{0ex}}^{3}\mathrm{He}{X}^{\ensuremath{-}}$ molecular anions are roughly three times as large as than that into $l=0\phantom{\rule{4pt}{0ex}}^{3}\mathrm{He}{X}^{\ensuremath{-}}$ molecular anions for $T\ensuremath{\in}[0.01,10]$ mK, where $l$ denotes the two-body rotational quantum number. In addition, for a given product state, the TBR rates of the $^{3}\mathrm{He}{+}^{3}\mathrm{He}+{\mathrm{H}}^{\ensuremath{-}}$ system are larger than that of the $^{3}\mathrm{He}+^{3}\mathrm{He}+{\mathrm{D}}^{\ensuremath{-}}$ system by roughly two orders of magnitude which could be ascribed to the major nonadiabatic couplings between the entrance and recombination channels.