Catalan's constant G is the alternating sum of reciprocals of odd squares: 1/1² − 1/3² + 1/5² − 1/7² + ⋯. It was studied by Eugène Catalan in 1865, though it appeared in earlier work. It has been computed to over 600 billion decimal places.
What makes G famous is not its value but its mystery: nobody knows if it is irrational. Almost all mathematicians believe it is: and probably transcendental: but no proof exists. It is one of the most embarrassing open questions in mathematics: a constant we can compute to arbitrary precision but cannot characterise.
G appears in physics (quantum field theory, the Ising model), combinatorics, and dozens of definite integrals. It equals −∫₀¹ ln(t)/(1+t²) dt, which can be verified numerically but not simplified to known constants.