[Abridged] We present the first results from a 1.1 mm confusion-limited map of the GOODS-S field taken with AzTEC on the ASTE telescope. We imaged a 270 sq. arcmin field to a 1\sigma depth of 0.48 - 0.73 mJy/beam, making this one of the deepest blank-field surveys at mm-wavelengths ever achieved. Although our GOODS-S map is extremely confused, we demonstrate that our source identification and number counts analyses are robust, and the techniques discussed in this paper are relevant for other deeply confused surveys. We find a total of 41 dusty starburst galaxies with S/N >= 3.5 within this uniformly covered region, where only two are expected to be false detections. We derive the 1.1mm number counts from this field using both a "P(d)" analysis and a semi-Bayesian technique, and find that both methods give consistent results. Our data are well-fit by a Schechter function model with (S', N(3mJy), \alpha) = (1.30+0.19 mJy, 160+27 (mJy/deg^2)^(-1), -2.0). Given the depth of this survey, we put the first tight constraints on the 1.1 mm number counts at S(1.1mm) = 0.5 mJy, and we find evidence that the faint-end of the number counts at S(850\mu m) < 2.0 mJy from various SCUBA surveys towards lensing clusters are biased high. In contrast to the 870 \mu m survey of this field with the LABOCA camera, we find no apparent under-density of sources compared to previous surveys at 1.1 mm. Additionally, we find a significant number of SMGs not identified in the LABOCA catalogue. We find that in contrast to observations at wavelengths < 500 \mu m, MIPS 24 \mu m sources do not resolve the total energy density in the cosmic infrared background at 1.1 mm, demonstrating that a population of z > 3 dust-obscured galaxies that are unaccounted for at these shorter wavelengths potentially contribute to a large fraction (~2/3) of the infrared background at 1.1 mm.