Capturing the potentially strong dependence among the peak concentrations of multiple air pollutants across a spatial region is crucial for assessing the related public health risks. In order to investigate the multivariate spatial dependence properties of air pollution extremes, we introduce a new class of multivariate max-stable processes. Our proposed model admits a hierarchical tree-based formulation, in which the data are conditionally independent given some latent nested positive stable random factors. The hierarchical structure facilitates Bayesian inference and offers a convenient and interpretable characterization. We fit this nested multivariate max-stable model to the maxima of air pollution concentrations and temperatures recorded at a number of sites in the Los Angeles area, showing that the proposed model succeeds in capturing their complex tail dependence structure.