  We consider a class of strongly edge-reinforced random walks, where the
corresponding reinforcement weight function is nondecreasing. It is known, from
Limic and Tarr\`{e}s [Ann. Probab. (2007), to appear], that the attracting edge
emerges with probability 1 whenever the underlying graph is locally bounded. We
study the asymptotic behavior of the tail distribution of the (random) time of
attraction. In particular, we obtain exact (up to a multiplicative constant)
asymptotics if the underlying graph has two edges. Next, we show some
extensions in the setting of finite graphs, and infinite graphs with bounded
degree. As a corollary, we obtain the fact that if the reinforcement weight has
the form $w(k)=k^{\rho}$, $\rho>1$, then (universally over finite graphs) the
expected time to attraction is infinite if and only if
$\rho\leq1+\frac{1+\sqrt{5}}{2}$.
