That's not a "base". In base-n, the valid digits are 0..[n-1].
In your system, 1 = 10 = 100 = 1000, and 11 == 101 = 110, and 101 = 1+1 = 11. It's useless as a notation for arithmetic resembling like what bases are used for.
In base 1, the only valid digit is 0, so the only valid number is unit.
Right, I addressed that in my post...
Well, maybe not "just like", the digit I have to use to represent each power of 1 is 1 and not 0, but I think that's a small concession.