Egyptian fraction representations of real numbers
There's a little bit of info here:
H. E. Salzer, The Approximation of Numbers as Sums of Reciprocals,
American Mathematical Monthly, Vol. 54, No. 3 (Mar., 1947), pp. 135-142. There are very few "naturally-occcurring" real numbers for which their expansion is known. One example where it is known is for (3-sqrt(5))/2, where the denominators are given by the (2^n)th Fibonacci number.