From natural ecology1-4 to clinical therapy5-8, cells are often exposed to mixtures of multiple drugs. Two competing null models are used to predict the combined effect of drugs: response additivity (Bliss) and dosage additivity (Loewe)9-11. Here, noting that these models diverge with increased number of drugs, we contrast their predictions with growth measurements of four phylogenetically distant microorganisms including Escherichia coli, Staphylococcus aureus, Enterococcus faecalis and Saccharomyces cerevisiae, under combinations of up to ten different drugs. In all species, as the number of drugs increases, Bliss maintains accuracy while Loewe systematically loses its predictive power. The total dosage required for growth inhibition, which Loewe predicts should be fixed, steadily increases with the number of drugs, following a square-root scaling. This scaling is explained by an approximation to Bliss where, inspired by R. A. Fisher's classical geometric model12, dosages of independent drugs add up as orthogonal vectors rather than linearly. This dose-orthogonality approximation provides results similar to Bliss, yet uses the dosage language as in Loewe and is hence easier to implement and intuit. The rejection of dosage additivity in favour of effect additivity and dosage orthogonality provides a framework for understanding how multiple drugs and stressors add up in nature and the clinic.