We report the Dzyaloshinskii-Moriya interaction (DMI) in Pt/Co/Re films. By measuring the current-induced hysteresis loop shift, we find that the interfacial DMI in the Pt/Co/Re structure is 2.1 pJ/m, which is stronger than that in Ir/Co/Pt asymmetric multilayers. The large DMI in this system can be attributed to an additive DMI at the Pt/Co and Co/Re interfaces; the Co/Re interface hosts a large DMI whose sign is the same as that at the Pt/Co interface. The additive DMI due to the large DMI at the Co/Re interface is consistent with theoretical predictions. The result provides a way to control the formation of chiral spin textures, such as magnetic skyrmions and chiral domain walls.