Question

The discovery of Transcendental numbers, or numbers that are not the root of any finite polynomial with rational coefficients.
Also, the proof that e and π were transcendental, the latter via the proof that e^a is only algebraic for transcendental values of a (and e^*i*π = -1 is algebraic, as is i, so therefore π must be transcendental). Their discovery, as well as the first explicitly created example, the Liouville number, sparked what's called "Transcendence theory".
And as it turns out, any randomly chosen real number is "almost surely" transcendental. In other words, the density of transcendental numbers among the real numbers is 1!!