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Ionic liquid (IL) electrolytes and carbon nanotube (CNT) electrodes have exhibited promising electrochemical performance in supercapacitors. Nevertheless, the adaptability of tricationic ILs (TILs) in CNT-based supercapacitors remains unknown. Herein, the performance of supercapacitors with (6,6), (8,8), (12,12), and (15,15) CNT electrodes in the TIL [C6(mim)3](Tf2N)3 was assessed via molecular dynamics simulations, paying attention to the electric double-layer (EDL) structures and the relations between the CNT curvature and capacitance. The results disclose that counterion and co-ion number densities near CNT electrodes have a marked reduction, compared with that of the graphene electrode. The capacitance of the EDL in the TIL increases significantly as the CNT curvature increases and the capacitance of the TIL/CNT systems is higher than that of the TIL/graphene system. Moreover, different EDL structures in the TIL and the monocationic IL (MIL) [C6mim][Tf2N] near CNT electrodes were revealed, showing higher-concentration anions [Tf2N]− at the CNT surfaces in the TIL. It is also verified that the TIL has a greater energy-storage ability under high potentials. Furthermore, the almost flat or weakly camel-like capacitance–voltage (C–V) curve of EDLs in the TIL turns into a bell shape in the MIL, because of the ion accumulation at the CNT surfaces and the associations between ions.