We obtain total galaxy X-ray luminosities, $L_X$, originating from individually detected point sources in a sample of 47 galaxies in 15 compact groups of galaxies (CGs). For the great majority of our galaxies, we find that the detected point sources most likely are local to their associated galaxy, and are thus extragalactic X-ray binaries (XRBs) or nuclear active galactic nuclei (AGNs). For spiral and irregular galaxies, we find that, after accounting for AGNs and nuclear sources, most CG galaxies are either within the $pm1sigma$ scatter of the Mineo et al. (2012) $L_X$ - star formation rate (SFR) correlation or have higher $L_X$ than predicted by this correlation for their SFR. We discuss how these excesses may be due to low metallicities and high interaction levels. For elliptical and S0 galaxies, after accounting for AGNs and nuclear sources, most CG galaxies are consistent with the Boroson et al. (2011) $L_X$ - stellar mass correlation for low-mass XRBs, with larger scatter, likely due to residual effects such as AGN activity or hot gas. Assuming non-nuclear sources are low- or high-mass XRBs, we use appropriate XRB luminosity functions to estimate the probability that stochastic effects can lead to such extreme $L_X$ values. We find that, although stochastic effects do not in general appear to be important, for some galaxies there is a significant probability that high $L_X$ values can be observed due to strong XRB variability.
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