That’s not how it’s works. Being “infinite” is not enough, the number 1.110100100010000… is “infinite”, without repeating patterns and dosen’t have other digits that 1 or 0.
If it’s infinite without repeating patterns then it just contain all patterns, no? Eh i guess that’s not how that works, is it? Half of all patterns is still infinity.
In some encoding scheme, those digits can represent something other than binary digits. If we consider your string of digits to truly be infinite, some substring somewhere will be meaningful.
to be fair, though, 1 and 0 are just binary representations of values, same as decimal and hexadecimal. within your example, we’d absolutely find the entire works of shakespeare encoded in ascii, unicode, and lcd pixel format with each letter arranged in 3x5 grids.