  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.
