  We consider a non-linear vertex-reinforced jump process (VRJP($w$)) on
$\mathbb{Z}$ with an increasing measurable weight function $w:[1,\infty)\to
[1,\infty)$ and initial weights equal to one. Our main goal is to study the
asymptotic behaviour of VRJP($w$) depending on the integrability of the
reciprocal of $w$. In particular, we prove that if $\int_1^{\infty}
\frac{\text{d}u}{w(u)} =\infty$ then the process is recurrent, i.e. it visits
each vertex infinitely often and all local times are unbounded. On the other
hand, if $\int_1^{\infty} \frac{\text{d} u}{w(u)} <\infty$ and there exists a
$\rho>0$ such that $t \mapsto w(t)^{\rho}\int_t^{\infty}\frac{\text{d}u}{w(u)}$
is non-increasing then the process will eventually get stuck on exactly three
vertices, and there is only one vertex with unbounded local time. We also show
that if the initial weights are all the same, VRJP on $\mathbb{Z}$ cannot be
transient, i.e. there exists at least one vertex that is visited infinitely
often. Our results extend the ones previously obtained by Davis and Volkov
[Probab. Theory Relat. Fields (2002)] who showed that VRJP with linear
reinforcement on $\mathbb{Z}$ is recurrent.
