Let $s$ be an infinite string of decimal digits, for example: \begin{array}{cccccccccc} s = 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 & \cdots \end{array} Consider a marker, the head, pointing to the first digit, $3$ in the above example. Interpret the digit under the head as an instruction to move the head $3$ digits to the right, i.e., to the $4$th digit. Now the head is pointing to $1$. Interpret this as an instruction to move $1$ place to the left. Continue in this manner, hopping through the string, alternately moving right and left. Think of the head as akin to the head of a Turing machine, and $s$ as the tape of instructions.
There are three possible behaviors. (1) The head moves off the left end of $s$:
\begin{array}{cccccccccc} 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} \end{array}
(2) The head goes into a cycle, e.g., when the head hits $0$:
\begin{array}{cccccccccccccc} 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} \\ 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 6 & 4 & 5 & 7 & 5 & 1 & 3 & 1 & 1 & 0 & 6 & 4 & 5 & 9 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} \end{array}
(3) The head moves off rightward to infnity:
\begin{array}{ccccccccccccc} 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 & 1 & 3 \\ \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{ ${}^{\wedge}$} & \text{} \\ \end{array}
This last string could be viewed as the decimal expansion of $31/99 = 0.3131313131313\cdots$.
Q1. What is an example of an irrational number $0.d_1 d_2 d_3 \cdots$ whose string $s=d_1 d_2 d_3 \cdots$ causes the head to hop rightward to infinity?
Q1.5. (Added). Is there an explicit irrational algebraic number with the hop-to-$\infty$ property?
I'm thinking of something like $\sqrt{7}-2$, the 2nd example above (which cycles).
Q2. More generally, which strings cause the head to hop rightward to infinity?
Update (summarizing answers, 13Apr2019). Q1. There are irrationals with the hop-to-$\infty$ property (@EthanBolker, @TheSimpliFire), but explicit construction requires using, e.g., the Thue-Morse sequence (@Wojowu). Q1.5. @EthanBolker suggests this may be difficult, and @Wojowu suggests it may be false (b/c: nine consecutive zeros): Perhaps no algebraic irrational has the hop-to-$\infty$ property. Q2. A partial algorithmic characterization by @TheSimpliFire.
$$ x 1^{x-2} y 1^{y-2} z1^{z-2} \ldots $$ 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.