After 500 steps, the value is the 53-digit number 39448887705043893375102470161238803295318090278129552 The question of whether such values can increase forever was considered by Eugène Catalan in 1887, and has remained unresolved since.

(The solution of the so-called dimer problem in 1961 also showed that for complete coverings of a square grid by 2-cell dominoes h = Catalan/( π Log[2]) ≃ 0.421 .)

The number of such strings containing 2n characters is the n th Catalan number Binomial[2n, n]/(n + 1) (as obtained from the generating function (1 - Sqrt[1 - 4x])/(2x) ).

(Catalan's Conjecture; 1844) • Can every integer not of the form 9n ± 4 be written as a 3 ± b 3 ± c 3 ?

Multiple integrals of rational functions can be more complicated, as in Integrate[1/(1 + x 2 + y 2 ), {x, 0, 1}, {y, 0, 1}]  HypergeometricPFQ[{1/2, 1, 1}, {3/2, 3/2}, 1/9]/6 + 1/2 π ArcSinh[1] - Catalan and presumably often cannot be expressed at all in terms of standard mathematical functions.