Strongly enhanced third-harmonic generation (THG) by the topological localization of an edge mode in a Su-Schrieffer-Heeger (SSH) chain of silicon photonic crystal nanocavities is demonstrated. The edge mode of the nanocavity chain not only naturally inherits resonant properties of the single nanocavity, but also exhibits the topological feature with mode robustness extending well beyond individual nanocavity. By engineering the SSH nanocavities with alternating strong and weak coupling strengths on a silicon slab, we observe the edge mode formation that entails a THG signal with three orders of magnitude enhancement compared with that in a trivial SSH structure. Our results indicate that the photonic crystal nanocavity chain could provide a promising on-chip platform for topology-driven nonlinear photonics.