We present Magellan/MIKE and Keck/HIRES high-resolution spectra of six red giant stars in the dwarf galaxy Segue 1. Including one additional Segue 1 star observed by Norris et al. (2010ApJ...722L.104N), high-resolution spectra have now been obtained for every red giant in Segue 1. Remarkably, three of these seven stars have metallicities below [Fe/H]=-3.5, suggesting that Segue 1 is the least chemically evolved galaxy known. We confirm previous medium-resolution analyses demonstrating that Segue 1 stars span a metallicity range of more than 2 dex, from [Fe/H]=-1.4 to [Fe/H]=-3.8. All of the Segue 1 stars are {alpha}-enhanced, with [{alpha}/Fe]~0.5. High {alpha}-element abundances are typical for metal-poor stars, but in every previously studied galaxy [{alpha}/Fe] declines for more metal-rich stars, which is typically interpreted as iron enrichment from supernova Ia. The absence of this signature in Segue 1 indicates that it was enriched exclusively by massive stars. Other light element abundance ratios in Segue 1, including carbon enhancement in the three most metal-poor stars, closely resemble those of metal-poor halo stars. Finally, we classify the most metal-rich star as a CH star given its large overabundances of carbon and s-process elements. The other six stars show remarkably low neutron-capture element abundances of [Sr/H]<-4.9 and [Ba/H]<-4.2, which are comparable to the lowest levels ever detected in halo stars. This suggests minimal neutron-capture enrichment, perhaps limited to a single r-process or weak s-process synthesizing event. Altogether, the chemical abundances of Segue 1 indicate no substantial chemical evolution, supporting the idea that it may be a surviving first galaxy that experienced only one burst of star formation.