Coronal mass ejections (CMEs) are the largest type of eruptions on the Sun and the main driver of severe space weather at the Earth. In this study, we implement a force-free spheromak CME description within 3D magnetohydrodynamic simulations to parametrically evaluate successive interacting CMEs within a representative heliosphere. We explore CME–CME interactions for a range of orientations, launch time variations, and CME handedness and quantify their geo-effectiveness via the primary solar wind variables and empirical measures of the disturbance storm time index and subsolar magnetopause standoff distance. We show how the interaction of two moderate CMEs between the Sun and the Earth can translate into extreme conditions at the Earth and how CME–CME interactions at different radial distances can maximize different solar wind variables that induce different geophysical impacts. In particular, we demonstrate how the orientation and handedness of a given CME can have a significant impact on the conservation and loss of magnetic flux, and consequently Bz, due to magnetic reconnection with the interplanetary magnetic field. This study thus implicates the identification of CME chirality in the solar corona as an early diagnostic for forecasting geomagnetic storms involving multiple CMEs.