Comet C/2006 W3 (Christensen) remained outside a heliocentric distance (R h) of 3.1 au throughout its apparition, but it presented an exceptional opportunity to directly sense a suite of molecules released from its nucleus. The Cryogenic Infrared Echelle Spectrograph at ESO-VLT detected infrared emissions from the three "hypervolatiles" (CO, CH4, and C2H6) that have the lowest sublimation temperatures among species that are commonly studied in comets by remote sensing. Even at R h = 3.25 au, the production rate of each molecule exceeded those measured for the same species in a number of other comets, although these comets were observed much closer to the Sun. Detections of CO at R h = 3.25, 4.03, and 4.73 au constrained its post-perihelion decrease in production rate, which most likely dominated the outgassing. At 3.25 au, our measured abundances scaled as CO/CH4/C2H6 ≈ 100/4.4/2.1. The C2H6/CH4 ratio falls within the range of previously studied comets at R h < 2 au, while CO/CH4 is comparatively high and similar to in situ measurements from Rosetta at ~10 km from the nucleus of 67P/Churyumov-Gerasimenko conducted at a very similar R h (3.15 au). The independent detections of ${\rm{H}}{}_{2}{\rm{O}}$ (Herschel Space Observatory) and CO (this work) imply a coma abundance ${{\rm{H}}}_{2}{\rm{O}}/\mathrm{CO}\approx 20 \% $ in C/2006 W3 near R h = 5 au. All these measurements are of high value for constraining models of nucleus sublimation (plausibly CO-driven) beyond R h = 3 au, where molecular detections in comets are still especially sparse.