Why not just use some sort of unlimited BigNum implementation? Yeah, for small numbers it's still ~2x the size of just storing an int, depending on implementation (or it can be: "int unless MAX_VALUE, in which case bignum is stored somewhere else") and it might be slower to operate on... but, on the other hand, you are already storing and processing a full video for every such counter!

Edit: Now I realize that would mean Google couldn't have made this joke. But I am still not sure this was foreseen by Youtube devs from day one.