Let a ternary string (a sequence of $0$s, $1$s, and $2$s) be in the set S if no substring immediately repeats itself in the larger string.
For example:
$20101$ is not in S because the substring $01$ is immediately followed by another $01$.
$0102010$ is in S because there are no substrings that immediately repeat. However, it is impossible to extend this string by adding $0,1,$ or $2$ to the end of it, since $01020100$ has $0$ repeated, $01020101$ as $01$ repeated, and $01020102$ has $0102$ repeated.
After much work by hand, I have found the string $01020120210120102120$, which is in S, and it still seems possible to append even more numbers while keeping the string in S (although at this point it is rather painstaking without a computer program to do it automatically).
So, overall, does S contain arbitrarily large strings?
