This problem sounds like it would succumb to a little brute force, followed by induction. The brute force solves all cases <k, for some relatively small k, by which time it is obvious that no other approach will do as well, and then simply show that your approach continues doing better & better as N gets bigger. The brute force part could be an exhaustive search by a computer. At 01:49 PM 4/28/2012, Warren Smith wrote:

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.