Then I will jump in and disagree completely- this does not involve any property of the real numbers- it is entirely a matter of how we represent the real numbers in a "base 10 positional notation". If we were to use, say, base 3, then "0.9999....= 1.0" would not be true (although, I imagine that "0.2222...= 1.0" would be). If we used some representation that was not a positional notation, the question would never arise.
This is a question about representation only, not about the real numbers. Indeed, 1 (and 0.9999....) are integers so, in particular, this has nothing to do with the "density" of the real numbers. (Whatever that means. The only "densities" I know of are relative to some set. Do you mean the fact that the rational numbers are dense in the real numbers?)