The Most Massive Galaxies at 3.0<z<4.0 in the NEWFIRM Medium-Band
  Survey: Properties and Improved Constraints on the Stellar Mass Function
  [Abridged] We use the NEWFIRM Medium-Band Survey (NMBS) to characterize the
properties of a mass-complete sample of 14 galaxies at 3.0<z<4.0 with
M_star>2.5x10^11 Msun, and to derive more accurate measurements of the
high-mass end of the stellar mass function (SMF) of galaxies at z=3.5, with
significantly reduced contributions from photometric redshift errors and cosmic
variance to the total error budget of the SMF. The typical very massive galaxy
at z=3.5 is red and faint in the observer's optical, with median r=26.1, and
rest-frame U-V=1.6. About 60% of the sample have optical colors satisfying
either the U- or the B-dropout color criteria, although ~50% of these galaxies
have r>25.5. About 30% of the sample has SFRs from SED modeling consistent with
zero. However, >80% of the sample is detected at 24 micron, with total infrared
luminosities in the range (0.5-4.0)x10^13 Lsun. This implies the presence of
either dust-enshrouded starburst activity (with SFRs of 600-4300 Msun/yr)
and/or highly-obscured active galactic nuclei (AGN). The contribution of
galaxies with M_star>2.5x10^11 Msun to the total stellar mass budget at z=3.5
is ~8%. We find an evolution by a factor of 2-7 and 3-22 from z~5 and z~6,
respectively, to z=3.5. The previously found disagreement at the high-mass end
between observed and model-predicted SMFs is now significant at the 3sigma
level. However, systematic uncertainties dominate the total error budget, with
errors up to a factor of ~8 in the densities, bringing the observed SMF in
marginal agreement with the predicted SMF. Additional systematic uncertainties
on the high-mass end could be introduced by either 1) the intense
star-formation and/or the very common AGN activities as inferred from the MIPS
24 micron detections, and/or 2) contamination by a significant population of
massive, old, and dusty galaxies at z~2.6.
