Does every irrational number contain “$666$” in its decimal expansion

Question

Let $x\in\mathbb{R}$. Say $x$ is satanic if $x$ contains "$666$" somewhere in its decimal expansions. Let $S$ be the set of such numbers. Is
$$
\mathbb{R} \setminus \mathbb{Q} \subseteq S
$$
More generally, for any finite string of digits, say $y = y_1y_2y_3\ldots y_{n-1}y_n$, does $x\in \mathbb{R}\setminus\mathbb{Q}$ necessarily contain $y$ in its decimal expansion.

It seems intuitive that, at least, for any $y$, $y$ should be contained in $\textit{almost every }$ irrational number, since the irrational numbers have an infinite decimal expansion. But, I could envision there exists some irrational number that follows some pattern such that it doesn't contain some $y$.

Answer 1

No. There are even transcendental numbers all of whose digits are $0$ and $1$.