Hopping to infinity along a string of digits
$$ x 1^{x-2} y 1^{y-2} z1^{z-2} \ldots $$ moves off to infinity for any sequence of digits $xyz\ldots$ between $3$ and $9$. Select a sequence that defines an irrational number.
More generally
$$ x 1 ?^{x-1} y 1 ?^{y-1} z 1 ?^{z-1} \ldots $$ works, where $?^n$ is an arbitrary string of $n$ digits, since those spots will never be hopped on.