Large-scale surveys over the last years have revealed about 300 QSOs at redshift above 6. Follow-up observations identified surprising properties, such as the very high black hole (BH) masses, spatial correlations with surrounding cold gas of the host galaxy, or high CIV-MgII velocity shifts. In particular, the discovery of luminous high-redshift quasars suggests that at least some black holes likely have large masses at birth and grow efficiently. We aim at quantifying quasar pairs at high redshift for a large sample of objects. This provides a new key constraint on a combination of parameters related to the origin and assembly for the most massive black holes: BH formation efficiency and clustering, growth efficiency and relative contribution of BH mergers. We observed 116 spectroscopically confirmed QSOs around redshift 6 with the simultaneous 7-channel imager GROND in order to search for companions. Applying identical colour-colour cuts as for those which led to the spectroscopically confirmed QSO, we perform LePHARE fits to the 26 best QSO pair candidates, and obtained spectroscopic observations for 11 of those. e do not find any QSO pair with a companion brighter than M1450(AB) < -26 mag within our 0.1-3.3 h^-1 cMpc search radius, in contrast to the serendipitous findings in the redshift range 4--5. However, a low fraction of such pairs at this luminosity and redshift is consistent with indications from present-day cosmological-scale galaxy evolution models. In turn, the incidence of L- and T-type brown dwarfs which occupy a similar colour space as z ~ 6 QSOs, is higher than expected, by a factor of 5 and 20, respectively.