  We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each
edge $\{x,x+1\}$ has the initial weight $x^{\alpha} \vee 1$, and each time an
edge is traversed, its weight is increased by $\Delta$. It is known that the
walk is recurrent if and only if $\alpha \leq 1$. The aim of this paper is to
study the almost sure behavior of the walk in the recurrent regime. For
$\alpha<1$ and $\Delta>0$, we obtain a limit theorem which is a counterpart of
the law of the iterated logarithm for simple random walks. This reveals that
the speed of the walk with $\Delta>0$ is much slower than $\Delta=0$. In the
critical case $\alpha=1$, our (almost sure) bounds for the trajectory of the
walk shows that there is a phase transition of the speed at $\Delta=2$.
