There is a simple intuitive explanation for how an "infinite" game is possible:
We can define two different sequences of three characters that start with 0 and end with 1: 001 and 011. Because they each start and end with a different character, we can never create a series of three characters by chaining two of these sequences.
Now we can go one step deeper and encode the "001" sequence as 0, and the "011" sequence as 1. We can generate our 001 and 011 pattern again, but with our encoded versions of 0 and 1, giving us these sequences: 001001011 and 001011011. These sequences again have the same characteristics, they start and end with different sub-sequences (001 and 011) so they can be chained without creating a series of three sub-sequences.
We can now use these larger sequences and encode these as 0 and 1, and so forth ad infinitum.
I guess it also hints at why the sequences keep getting smaller when you compress it several times (as mentioned in a footnote). Each compression peels away layers in this expansion. That’s my intuition at least.
We can define two different sequences of three characters that start with 0 and end with 1: 001 and 011. Because they each start and end with a different character, we can never create a series of three characters by chaining two of these sequences.
Now we can go one step deeper and encode the "001" sequence as 0, and the "011" sequence as 1. We can generate our 001 and 011 pattern again, but with our encoded versions of 0 and 1, giving us these sequences: 001001011 and 001011011. These sequences again have the same characteristics, they start and end with different sub-sequences (001 and 011) so they can be chained without creating a series of three sub-sequences.
We can now use these larger sequences and encode these as 0 and 1, and so forth ad infinitum.