CR7 is the brightest z = 6.6 Ly ... emitter (LAE) known to date, and spectroscopic follow-up by Sobral et al. suggests that CR7 might host Population (Pop) III stars. We examine this interpretation using cosmological hydrodynamical simulations. Several simulated galaxies show the same 'Pop III wave' pattern observed in CR7. However, to reproduce the extreme CR7 Ly.../He ii1640 line luminosities ($L_...\rm \alpha /He\,\small ...II...$) a top-heavy initial mass function and a massive ( ... 10... M...) Pop III burst with age ... 2 Myr are required. Assuming that the observed properties of Ly ... and He ii emission are typical for Pop III, we predict that in the COSMOS/UDS/SA22 fields, 14 out of the 30 LAEs at z = 6.6 with L... > 10... erg s... should also host Pop III stars producing an observable $L_...\rm He\,\small ...II...\gtrsim 10 greater than or equal to ..42.7...\,...\rm erg...\,...\rm s... greater than or equal to ..-1...$. As an alternate explanation, we explore the possibility that CR7 is instead powered by accretion on to a direct collapse black hole. Our model predicts L..., $L_...\rm He\,\small ...II...$, and X-ray luminosities that are in agreement with the observations. In any case, the observed properties of CR7 indicate that this galaxy is most likely powered by sources formed from pristine gas. We propose that further X-ray observations can distinguish between the two above scenarios. (ProQuest: ... denotes formulae/symbols omitted.)