  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 strictly
increasing, then the process visits only a finite number of vertices. In the
case where $w$ is asymptotically 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.
