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 as follows:
The first digits of are . After this string of 's terminates, the next digits are exactly the digits of .
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.