Find an irrational number whose binary development does not contain strings of $0$s or $1$s of arbitrary length 
Find an irrational number whose binary development does not contain strings of $0$s or $1$s of arbitrary length.

I found it hard to find such an example since the only two digits we can use are $0$ and $1$. How can we construct such an example?
 A: Consider $0.\overline{10}_2$ and change every $0$ in position $(2n)!$ after the dot with a $1$.
Such binary representation is aperiodic (so it gives an irrational number) and given by a concatenation of $0$, $1$ and $111$ only. Additionally, such number is trascendental, as the sum between $\frac{2}{3}$ and a Liouville number.
A: Call $a=101$, $b=1001$. Using only these two strings as "digits" write a non-repeating infinite pattern, for example
$$0.abaabaaabaaaab...$$
and then substitute $a=101$, $b=1001$.
A: The following binary expansion contains no substring $11$ or longer and no substring $000$ or longer:
$$0.01\color{crimson}00101\color{crimson}0010101\color{crimson}001010101\color{crimson}0\ldots$$
It’s not eventually periodic, so it’s irrational.
A: Take an irrational number (e.g. $\sqrt 2$) in binary format, and replace each $1$ with the sequence $011$ and each $0$ with the sequence $001$. Then you don't have more than two consecutive $0$'s or $1$'s and the generated number is also irrational (does not have periodic substrings).
A: The number
$$.1?0?1?0?1?0?\dots$$
contains no string of $0$s or $1$s of length $4$. There are uncountably many ways to give values to the $?$ marks, most of them resulting in irrational numbers.
Better, the Thue–Morse sequence represents an irrational number and has no strings of length $3$:
$$.0110100110010110\dots$$
Or the Kolakoski sequence (with $2$s changed to $0$s):
$$.10011010010011011\dots$$
