The trouble with rationals is that the denominators keep growing.
Yes and no. It is true that there are cases where they do grow (but do you often add numbers like 132511/43 and 27/67 and 1024/853?), but there are many more cases where they don't. If you add or subtract many numbers in decimal format which most of us use everyday (monetary amount, physical measures made in the metric system, etc.) with, say two to five decimal places (or more), and the denominators will not grow and will often be a power of 10 (possibly multiplied by a power of 2 or a power of 5, something that can easily be brought back to a power of 10 by adjusting the numerator by the same factor). So in such cases, you'll never reach the 64-bit limit.