We present new determinations of the iron abundance for 220 stars belonging to the stellar system Terzan 5 in the Galactic bulge. The spectra have been acquired with FLAMES at the Very Large Telescope of the European Southern Observatory and DEIMOS at the Keck II Telescope. This is by far the largest spectroscopic sample of stars ever observed in this stellar system. From this dataset, a subsample of targets with spectra unaffected by TiO bands was extracted and statistically decontaminated from field stars. Once combined with 34 additional stars previously published by our group, a total sample of 135 member stars covering the entire radial extent of the system has been used to determine the metallicity distribution function of Terzan 5. The iron distribution clearly shows three peaks: a super-solar component at [Fe/H]$≃0.25$ dex, accounting for 29% of the sample, a dominant sub-solar population at [Fe/H]$≃-0.30$ dex, corresponding to 62% of the total, and a minor (6%) metal-poor component at [Fe/H]$≃-0.8$ dex. Such a broad, multi-modal metallicity distribution demonstrates that Terzan 5 is not a genuine globular cluster but the remnant of a much more complex stellar system. ; Comment: 29 pages, 10 figures. Accepted for publication by ApJ