The irrational constant $R = e^{\pi \sqrt{163}}$, which is very close to an integer. The name "Ramanujan's constant" derives from an April Fool's joke played by Martin Gardner, where he claimed that this number is an integer and Ramanujan conjectured that in 1914.
How they were obtained: Checked here against exp(pi*sqrt(163)) computed in ball arithmetic at 4000 bits, which covers every stored digit rather than the first hundred: the stored value is within one unit in the last place of the 1000 held.
