We present deep observations at 450 and 850 μm in the Extended Groth Strip field taken with the SCUBA-2 camera mounted on the James Clerk Maxwell Telescope as part of the deep SCUBA-2 Cosmology Legacy Survey (S2CLS), achieving a central instrumental depth of σ450 = 1.2 mJy beam−1 and σ850 = 0.2 mJy beam−1. We detect 57 sources at 450 μm and 90 at 850 μm with signal-to-noise ratio >3.5 over ∼70 arcmin2. From these detections, we derive the number counts at flux densities S450 > 4.0 mJy and S850 > 0.9 mJy, which represent the deepest number counts at these wavelengths derived using directly extracted sources from only blank-field observations with a single-dish telescope. Our measurements smoothly connect the gap between previous shallower blank-field single-dish observations and deep interferometric ALMA results. We estimate the contribution of our SCUBA-2 detected galaxies to the cosmic infrared background (CIB), as well as the contribution of 24 μm-selected galaxies through a stacking technique, which add a total of 0.26 ± 0.03 and 0.07 ± 0.01 MJy sr−1, at 450 and 850 μm, respectively. These surface brightnesses correspond to 60 ± 20 and 50 ± 20 per cent of the total CIB measurements, where the errors are dominated by those of the total CIB. Using the photometric redshifts of the 24 μm-selected sample and the redshift distributions of the submillimetre galaxies, we find that the redshift distribution of the recovered CIB is different at each wavelength, with a peak at z ∼ 1 for 450 μm and at z ∼ 2 for 850 μm, consistent with previous observations and theoretical models.