The source counts of galaxies discovered at submillimetre and millimetre wavelengths provide important information on the evolution of infrared-bright galaxies. We combine the data from six blank-field surveys carried out at 1.1 mm with AzTEC, totalling 1.6 deg2 in area with root-mean-square depths ranging from 0.4 to 1.7 mJy, and derive the strongest constraints to date on the 1.1 mm source counts at flux densities S1100= 1–12 mJy. Using additional data from the AzTEC Cluster Environment Survey to extend the counts to S1100∼ 20 mJy, we see tentative evidence for an enhancement relative to the exponential drop in the counts at S1100∼ 13 mJy and a smooth connection to the bright source counts at >20 mJy measured by the South Pole Telescope; this excess may be due to strong-lensing effects. We compare these counts to predictions from several semi-analytical and phenomenological models and find that for most the agreement is quite good at flux densities ≳ 4 mJy; however, we find significant discrepancies (≳ 3σ) between the models and the observed 1.1-mm counts at lower flux densities, and none of them is consistent with the observed turnover in the Euclidean-normalized counts at S1100≲ 2 mJy. Our new results therefore may require modifications to existing evolutionary models for low-luminosity galaxies. Alternatively, the discrepancy between the measured counts at the faint end and predictions from phenomenological models could arise from limited knowledge of the spectral energy distributions of faint galaxies in the local Universe.