We present the detection of a sample of galaxies which are extremely faint in the optical and near-infrared but bright at mid-infrared wavelengths. This population of galaxies, missed by the deepest HST surveys such as CANDELS or Hubble Frontier Fields, are considerably bright in IRAC. The bulk of the sample (65%) is located in a 2σ region around the main sequence. Approximately 20% of the BBGs are very dusty starbursts with strong mid-to-far infrared detections and extreme star formation rates. The remaining, 15%, are located more than 2σ below the main sequence and might be either regular star-forming galaxies or quiescent systems. Nearly one third of them are MIPS emitters, most probably revealing the presence of an obscured AGN co-existing with the intense star formation, as measured by Herchel. Only 2 sources are detected in X-rays and 3 at sub-millimiter or radio wavelengths. Our results point out that BBGs significantly contribute (35%) to the general population of massive red galaxies at z=4-6 and that one of every ten massive log(M/M_⊙)>11 galaxies in the local Universe was assembled in the first 1.5 Gyr after the Big Bang.