By the Cantor slash argument, as explained for example here (at about 4:00), a new real number can always be generated out of any list of real number decimal expansions by taking the digits along the diagonal of the list and calling those digits a new number.
I am having a hard time understanding why Cantor's slash argument does not also apply to the rationals.
The rationals do not have infinite non-repeating decimal expansions, but they can be made arbitrarily large. Therefore we can always make list of rationals like the one you make for reals in the slash argument, with the one difference that the "..." at the end of each number is interpreted as "goes on to become arbitrarily large" instead of "goes on forever." On any such list, it will always be possible to generate a new number that is not on the list by taking the diagonals in the same way you do for reals. The rationals are thus unlistable or uncountable in the same way as the reals are. (Please point out the flaw in this reasoning.)
If I may also pose a very closely related follow-up:
If the Ford Circle algorithm is allowed to run infinitely, it will, in that limit, completely fill up every possible spot on the number line with rational numbers. Where on the number line, then, in that limit, is there any space for irrational and transcendental numbers? Or what am I missing here?