Because that’s what happens when you multiply something by 10: 10 x 123 = 1230.
Obviously, this isn’t the rule, it’s just a convenient short-cut for deriving the result. The rule isn’t: shift all the digits to the left and fill the last place with a 0. The rule is: add the multiplicand to itself a number of times equal to the multiplier. So with 10 x 123, you would add 123 to itself 10 times. You start with 123, then get 246, then 369, etc. until we get 1230. Nowhere in that process are we shifting digits to the left. It’s just convenient that, when multiplying by 10, we so happen to get a result that can be derived by a much simpler short-cut: shift the digits to the left and tack on a 0.
This short-cut applies to all numbers with finite decimal expansions, but because it is not the actual rule of multiplication, we cannot necessarily say it carries over to numbers with infinite decimal expansion. We’d have to derive a thorough explanation for why we get this short-cut when multiplying numbers with finite decimal expansions by 10, and then see if that explanation carries over to numbers with infinite decimal expansions.
So instead of asking whether (10 * 0.\dot9 = 9.\dot9), we ask whether (10 * (3 * 0.\dot3) = 10 * (3 * \frac{1}{3}) = 9.\dot9)?
I doubt that would convince the skeptics. Magnus has already ruled out the fact that (\frac{1}{3} = 0.\dot3).
It’s a tangent. This thread is full of tangents. Personally, I’m okay with that as I’m not actually trying to get to the bottom of the main question–does (0.\dot9) really equal 1?–I just enjoy a good debate regardless of where it leads.
(But just to address your point, if there is a largest number, it would fundamentally change the way we understand numbers (it would for me at least), and this could affect the way we understand the question of does (0.\dot9) = 1.)