TITLE: Hopping to infinity along a string of digits
QUESTION [10 upvotes]: 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.

REPLY [12 votes]: $$
x 1^{x-2} y 1^{y-2} z1^{z-2} \ldots
$$
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.