Context. Since low-mass stars have low luminosities, orbits at which liquid water can exist on Earth-sized planets are relatively close-in, which produces Doppler signals that are detectable using state-of-the-art Doppler spectroscopy. Aims. GJ 667C is already known to be orbited by two super-Earth candidates. We have recently applied developed data analysis methods to investigate whether the data supports the presence of additional companions. Methods. We obtain new Doppler measurements from HARPS extracted spectra and combined them with those obtained from the PFS and HIRES spectrographs. We used Bayesian and periodogram-based methods to re-assess the number of candidates and evaluated the confidence of each detection. Among other tests, we validated the planet candidates by analyzing correlations of each Doppler signal with measurements of several activity indices and investigated the possible quasi-periodic nature of signals. Results. Doppler measurements of GJ 667C are described better by six (even seven) Keplerian-like signals: the two known candidates (b and c); three additional few-Earth mass candidates with periods of 92, 62, and 39 days (d, e and f); a cold super-Earth in a 260-day orbit (g) and tantalizing evidence of a ∼1 M_⊙ object in a close-in orbit of 17 days (h). We explore whether long-term stable orbits are compatible with the data by integrating 8 × 10⁴ solutions derived from the Bayesian samplings. We assess their stability using secular frequency analysis. Conclusions. The system consisting of six planets is compatible with dynamically stable configurations. As for the solar system, the most stable solutions do not contain mean-motion resonances and are described well by analytic Laplace-Lagrange solutions. Preliminary analysis also indicates that masses of the planets cannot be higher than twice the minimum masses obtained from Doppler measurements. The presence of a seventh planet (h) is supported by the fact that it appears squarely centered on the only island of stability left in the six-planet solution. Habitability assessments accounting for the stellar flux, as well as tidal dissipation effects, indicate that three (maybe four) planets are potentially habitable. Doppler and space-based transit surveys indicate that 1) dynamically packed systems of super-Earths are relatively abundant and 2) M-dwarfs have more small planets than earlier-type stars. These two trends together suggest that GJ 667C is one of the first members of an emerging population of M-stars with multiple low-mass planets in their habitable zones.