In an irrational number what is the longest possible sequence of a single digit that can occur?
Just by using the word "irrational" you automatically attach absolutely no general rules to the decimal expansion of the number. Here is a simple proof.
Define $\displaystyle x_n$ as follows:
The first $\displaystyle n$ digits of $\displaystyle x_n$ are $\displaystyle 1$. After this string of $\displaystyle 1$'s terminates, the next digits are exactly the digits of $\displaystyle \pi$.
$\displaystyle x_{20}=.111111111111111111113141592653589793238462 643383279502884197...$
This number is automatically irrational because it has an infinite non-repeating decimal expansion, and yet n can take on whatever value you choose, whether it be 20 or 20 trillion.