The dynamics of three three-dimensional repressilators globally coupled by a quorum sensing mechanism was numerically studied. This number (three) of coupled repressilators is sufficient to obtain such a set of self-consistent oscillation frequencies of signal molecules in the mean field that results in the appearance of self-organized quasiperiodicity and its complex evolution over wide areas of model parameters. Numerically analyzing the invariant curves as a function of coupling strength, we observed torus doubling, three torus arising via quasiperiodic Hopf bifurcation, the emergence of resonant cycles, and secondary Neimark-Sacker bifurcation. A gradual increase in the oscillation amplitude leads to chaotizations of the tori and to the birth of weak, but multidimensional chaos.