We use the SCUBA-2 submillimeter camera mounted on the JCMT to obtain extremely deep number counts at 450 and 850um. We combine data on two cluster lensing fields, A1689 and A370, and three blank fields, CDF-N, CDF-S, and COSMOS, to measure the counts over a wide flux range at each wavelength. We use statistical fits to broken power law representations to determine the number counts. This allows us to probe to the deepest possible level in the data. At both wavelengths our results agree well with the literature in the flux range over which they have been measured, with the exception of the 850um counts in CDF-S, where we do not observe the counts deficit found by previous single-dish observations. At 450um, we detect significant counts down to ~1mJy, an unprecedented depth at this wavelength. By integrating the number counts above this flux limit, we measure 113.9^{+49.7}_{-28.4} Jydeg^{-2} of the 450um extragalactic background light (EBL). The majority of this contribution is from sources with S_450um between 1-10mJy, and these sources are likely to be the ones that are analogous to the local luminous infrared galaxies (LIRGs). At 850um, we measure 37.3^{+21.1}_{-12.9} Jydeg^{-2} of the EBL. Because of the large systematic uncertainties on the COBE measurements, the percentage of the EBL we resolve could range from 48%-153% (44%-178%) at 450 (850)um. Based on high-resolution SMA observations of around half of the 4sigma 850um sample in CDF-N, we find that 12.5^{+12.1}_{-6.8}% of the sources are blends of multiple fainter sources. This is a low multiple fraction, and we find no significant difference between our original SCUBA-2 850um counts and the multiplicity corrected counts.
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