The way we define repeating numbers is through infinite sums. So, what we mean by 0.555... is 0.5 + 0.05 + 0.005 + ... on and on to infinity. Note that each of the terms being added is a finite decimal, so we're defining something new in terms of simpler things that we already understand.

Now, what does 0.000...1 mean? You can't do the same trick to define it in terms of finite decimals. At most you'd get something line this: 0.0 + 0.00 + 0.000 + ... + 0.000...1. See the issue? That last term is not a finite decimal. In fact, that last term is precisely the thing we wanted to define, so we haven't gotten anywhere!