We study asymptotic behaviours of a non-linear vertex-reinforced jump process defined on an arbitrary infinite graph with bounded degree. We prove that if the reinforcement function $w$ is reciprocally integrable and non-decreasing, then the process visits only a finite number of vertices. In the case where $w$ is approximately equal to a super-linear polynomial, we show that the process eventually gets stuck on a star-shaped subgraph and there is exactly one vertex with unbounded local time.