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abstract: 'We use the optical to mid-infrared coverage of the NEWFIRM Medium-Band Survey (NMBS) to characterize, for the first time, the properties of a mass-complete sample of 14 galaxies at $3.0 \leq z < 4.0$ with $M_{\rm star}>2.5\times 10^{11}$ M$_{\sun}$, and to derive significantly more accurate measurements of the high-mass end of the stellar mass function (SMF) of galaxies at $3.0 \leq z < 4.0$. The accurate photometric redshifts and well-sampled SEDs provided by the NMBS combined with the large surveyed area result in significantly reduced contributions from photometric redshift errors and cosmic variance to the total error budget of the SMF. The typical very massive galaxy at $3.0 \leq z < 4.0$ is red and faint in the observer’s optical, with a median $r$-band magnitude of $\langle r_{\rm tot}\rangle=26.1$, and median rest-frame $U-V$ colors of $\langle U-V\rangle=1.6$. About 60% of the mass-complete sample have optical colors satisfying either the $U$- or the $B$-dropout color criteria, although $\sim$50% of these galaxies have $r>25.5$. We find that $\sim$30% of the sample has SFRs from SED modeling consistent with zero, although SFRs of up to $\sim1-18$ M$_{\sun}$ yr$^{-1}$ are also allowed within 1 $\sigma$. However, $>80$% of the sample is detected 