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We find that the chemical abundances and dynamics of APOGEE and GALAH stars in the local stellar halo are inconsistent with a scenario in which the inner halo is primarily composed of debris from a single, massive, ancient merger event, as has been proposed to explain the Gaia-Enceladus/Gaia Sausage (GSE) structure. The data contains trends of chemical composition with energy which are opposite to expectations for a single massive, ancient merger event, and multiple chemical evolution paths with distinct dynamics are present. We use a Bayesian Gaussian mixture model regression algorithm to characterize the local stellar halo, and find that the data is best fit by a model with four components. We interpret these components as the VRM, Cronus, Nereus, and Thamnos; however, Nereus and Thamnos likely represent more than one accretion event because the chemical abundance distributions of their member stars contain many peaks. Although the Cronus and Thamnos components have different dynamics, their chemical abundances suggest they may be related. We show that the distinct low and high alpha halo populations from Nissen & Schuster (2010) are explained by VRM and Cronus stars, as well as some in-situ stars. Because the local stellar halo contains multiple substructures, different popular methods of selecting GSE stars will actually select different mixtures of these substructures, which may change the apparent chemodynamic properties of the selected stars. We also find that the Splash stars in the solar region are shifted to higher v_phi and slightly lower [Fe/H] than previously reported.