Narrowband-filter photometry of Comet 96P/Machholz 1 was obtained at Lowell Observatory during the comet's 2007 apparition. Production rates of OH, CN, C₂, C₃, and NH were derived from these data sets, and the quantity A(θ)fρ—a proxy measure of the dust production—was also calculated. Relative abundances, expressed as ratios of production rates with respect to OH (a measure of the water abundance), were compared to those measured in other comets. Comet Machholz 1 is shown to be depleted of CN by about a factor of 72 from average, while C₂ and C₃ are also low, but "only" by factors of 8 and 19, respectively, from "typical" composition (based on an update to the classifications by A'Hearn et al.). In contrast, NH is near the mid-to-upper end of its normal range. This extremely low CN-to-OH ratio for Machholz 1 indicates that it is either compositionally associated with Comet Yanaka (1988r; 1988 Y1), which was strongly depleted in CN and C₂, but not NH₂, or represents a new compositional class of comets, since Yanaka had a much greater depletion of C₂ (>100×) than did Machholz 1 (8×). Evidence strongly suggests that the extremely anomalous compositions of these two comets are primordial rather than from recent thermal processing. It remains unclear whether these comets formed at a location in our solar system with unusual conditions and a low probability of being gravitationally perturbed into the inner solar system, or if one or both objects are interstellar interlopers.