The longest sequence of digits that eventually repeat (as defined in the article) is unboundedly large for any infinite sequence of digits, not just pi!
Using the pigeonhole principle, there must be at least one length N repeating string in the first N(N!+1) characters of any string.
Using the pigeonhole principle, there must be at least one length N repeating string in the first N(N!+1) characters of any string.