We study the structure and dynamics of extreme flaring events on young stellar objects (YSOs) observed in hard X-rays by the Nuclear Spectroscopic Telescope Array (NuSTAR). During 2015 and 2016, NuSTAR made three observations of the star-forming region ρ Ophiuchi, each with an exposure ∼50 ks. NuSTAR offers unprecedented sensitivity above ∼7 keV, making this data set the first of its kind. Through improved coverage of hard X-rays, it is finally possible to directly measure the high-energy thermal continuum for hot plasmas and to sensitively search for evidence of nonthermal emission from YSO flares. During these observations, multiple flares were observed, and spectral and timing analyses were performed on three of the brightest flares. By fitting an optically thin thermal plasma model to each of these events, we found flare plasma heated to high temperatures (∼40-80 MK) and determined that these events are ∼1000 times brighter than the brightest flares observed on the Sun. Two of the studied flares showed excess emission at 6.4 keV, and this excess may be attributable to iron fluorescence in the circumstellar disk. No clear evidence for a nonthermal component was observed, but upper limits on nonthermal emission allow for enough nonthermal energy to account for the estimated thermal energy in the flare on protostar IRS 43, which is consistent with the standard model for solar and stellar flares.