We present near-IR photometry and spectroscopy of 30 extremely luminous radio and mid-IR selected galaxies. With bolometric luminosities exceeding $sim10^{13}$ $rm{L_{odot}}$ and redshifts ranging from $z = 0.880-2.853$, we use VLT instruments X-shooter and ISAAC to investigate this unique population of galaxies. Broad multi-component emission lines are detected in 18 galaxies and we measure the near-IR lines $rm{Hrm{beta}}$, $text{[OIII]}rm{lambda}rm{lambda}4959,5007$ and $rm{Hrm{alpha}}$ in six, 15 and 13 galaxies respectively, with 10 $rm{Lyalpha}$ and five CIV lines additionally detected in the UVB arm. We use the broad $text{[OIII]}rm{lambda}5007$ emission lines as a proxy for the bolometric AGN luminosity, and derive lower limits to supermassive black hole masses of $10^{7.9}$-$10^{9.4}$ $text{M}_{odot}$ with expectations of corresponding host masses of $10^{10.4}$-$10^{12.0}$ $text{M}_{odot}$. We measure $rm{lambda}_{Edd}$ > 1 for eight of these sources at a $2sigma$ significance. Near-IR photometry and SED fitting are used to compare stellar masses directly. We detect both Balmer lines in five galaxies and use these to infer a mean visual extinction of $A_{V}$ = 2.68 mag. Due to non-detections and uncertainties in our $rm{Hrm{beta}}$ emission line measurements, we simulate a broad $rm{Hrm{beta}}$ line of FWHM = 1480 $rm{kms^{-1}}$ to estimate extinction for all sources with measured $rm{Hrm{alpha}}$ emission. We then use this to infer a mean $A_{V}=3.62$ mag, demonstrating the highly-obscured nature of these galaxies, with the consequence of increasing our estimates of black-hole masses by an 0.5 orders of magnitude in the most extreme and obscured cases.
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