Given the Cayley graph of a group $G$ (for some fixed generating set $S$), consider the set $J_S$ of all the elements which lie on some *infinite* geodesic ray starting at the identity element of $G$, $e_G$.
Let $d$ denotes the distance in the Cayley graph

A simple example where $J_S \neq G$ is $G = \mathbb{Z} \times \mathbb{Z}_2$ with the generating set $S = \lbrace (\pm 1,0), (\pm 1 ,1) \rbrace$. In that case, $(0,1)$ does not lie on an infinite geodesic ray starting at $(0,0)$.

**Question:** Is there a choice of $G$, $S$ and $x \in G \setminus S$ so that $d(x,J_S) > \tfrac{1}{2} d(x,e_G)$.

[Edit: As David pointed out below, the interesting question would be: does $\limsup \frac{d(x_n,J_S)}{ d(x_n,e_G)} \leq \tfrac{1}{2}$ (where $x_n$ is some enumeration of the vertices)?

3more comments