We report Keck/ESI and VLT/UVES observations of three super-damped Lyman-alpha quasar absorbers with H I column densities log N(HI) >= 21.7 at redshifts z=2-2.5. All three absorbers show similar metallicities (-1.3 to -1.5 dex), and dust depletion of Fe, Ni, and Mn. Two of the absorbers show supersolar [S/Zn] and [Si/Zn]. We combine our results with those for other DLAs to examine trends between N(HI), metallicity, dust depletion. A larger fraction of the super-DLAs lie close to or above the line [X/H]=20.59-log N(HI) in the metallicity vs. N(HI) plot, compared to the less gas-rich DLAs, suggesting that super-DLAs are more likely to be rich in molecules. Unfortunately, our data for Q0230-0334 and Q0743+1421 do not cover H2 absorption lines. For Q1418+0718, some H2 lines are covered, but not detected. CO is not detected in any of our absorbers. For DLAs with log N(HI) < 21.7, we confirm strong correlation between metallicity and Fe depletion, and find a correlation between metallicity and Si depletion. For super-DLAs, these correlations are weaker or absent. The absorbers toward Q0230-0334 and Q1418+0718 show potential detections of weak Ly-alpha emission, implying star formation rates of about 1.6 and 0.7 solar masses per year, respectively (ignoring dust extinction). Upper limits on the electron densities from C II*/C II or Si II*/Si II are low, but are higher than the median values in less gas-rich DLAs. Finally, systems with log N(HI) > 21.7 may have somewhat narrower velocity dispersions delta v_90 than the less gas-rich DLAs, and may arise in cooler and/or less turbulent gas.
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