We analyze 1598 serendipitous Chandra X-ray observations of 462 radio-quiet quasars to constrain the frequency of extreme amplitude X-ray variability that is intrinsic to the quasar corona and innermost accretion flow. The quasars in this investigation are all spectroscopically confirmed, optically bright (m_i_<=20.2), and contain no identifiable broad absorption lines in their optical/ultraviolet spectra. This sample includes quasars spanning z~0.1-4 and probes X-ray variability on timescales of up to ~12 rest-frame years. Variability amplitudes are computed between every epoch of observation for each quasar and are analyzed as a function of timescale and luminosity. The tail-heavy distributions of variability amplitudes at all timescales indicate that extreme X-ray variations are driven by an additional physical mechanism and not just typical random fluctuations of the coronal emission. Similarly, extreme X-ray variations of low-luminosity quasars seem to be driven by an additional physical mechanism, whereas high-luminosity quasars seem more consistent with random fluctuations. The amplitude at which an X-ray variability event can be considered extreme is quantified for different timescales and luminosities. Extreme X-ray variations occur more frequently at long timescales ({DELTA}t>=300-days) than at shorter timescales, and in low-luminosity quasars compared to high-luminosity quasars over a similar timescale. A binomial analysis indicates that extreme intrinsic X-ray variations are rare, with a maximum occurrence rate of <2.4% of observations. Finally, we present X-ray variability and basic optical emission-line properties of three archival quasars that have been newly discovered to exhibit extreme X-ray variability.