A perhaps useful attack on a problem of this ilk is: Read http://en.wikipedia.org/wiki/E-function the function F(x)=exp(exp(x)-1) perhaps is an E-function, and indeed iterating the function G(X)=exp(x)-1 any fixed finite number of times, perhaps yields an E-function. If so, then by the Siegel–Shidlovsky theorem it would follow that if x is a nonzero integer then G(G(G(...x))) must be non-algebraic. To investigate whether this line of attack could work, I ran the following MAPLE9 script: foo := series( exp(exp(x)-1), x, 301 ); for k from 1 to 300 do print( k, evalf(log(coeff(foo,x,k)*k!),4) ); od; with the conclusion that k! times the series coefficient of x^k [coefficients indeed all are of the form integer/k!] seems to be growing too fast as a function of k to qualify for E-function status. ... Sorry. ---- A related conjecture I made a long time ago, which nobody has ever resolved, is this. Start with owning the numbers 0 and 1. Each step, with two numbers you already own (perhaps identical), compute A+B, A-B, A*B, A/B, or 2^A. CONJECTURE: the greatest finite real number reachable in N>0 steps is 2^2^2^...^2 with N-1 uses of "^" and evaluating right to left.