Here is another example of a number that was thought to be rational until it was proved to be irrational. Erdős conjectured that not much more integers are representable as a sum of two squareful numbers than as a sum of two squares. More precisely, he conjectured that up to $x$ the number of integers in the first set is $x/(\log x)^{1/2+o(1)}$. Blomer proved that the exponent $1/2$ is wrong, the correct value is $1-2^{-1/3}$. He also showed that the same estimate is valid for sums of a square and a squareful number. See J. London Math. Soc. (2) 71 (2005), 69-84.