[dacohenii]

I'm pretty sure it's a real number in that it falls within ℝ (the set of all real numbers).

[RHSeeger]

I doesn't, because 0.666....7 is not a valid representation for a number. The ... means goes on forever, and you cannot put something "after forever". It's not different than saying "0.0j" is a real number in base 10; it's not, that string does not represent a number in our number base 10 number system.

[hilbertseries]

There’s not a way to define it, typically you’d define .666... as the sum of 6/10^n from 1 to infinity. This decimal representation does not terminate, so you can’t put a 7 at the “end of it” because there is no end.