Michael Reid:

...

WDS: 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.

OK, in the first step, I compute  1 + 1 = 2 .  In the next step, compute  2^2 = 4  (which may as well be  2 + 2  or  2 * 2 ).

But in the third step you want me to do  2^4 , whereas I'd rather do  4^4 , since I already "own"  4 . Did I misunderstand something?

--the operation that is allowed is 2^x, not x^y.

Henry Brady: Brute force, do the first few cases exhaustive search, then "obvious" must continue

--actually, I did do some brute force on this to prove my conjecture for, I think it was, N<=6 or something. But the "obvious" part, I was never able to do. You might be able to generate two very close real numbers then 1/(A-B) is very large. I doubt it, but...

Schanuel's conjecture...

I would think proving e^e^e^79 is noninteger and proving Schanuel's conjecture would be approximately equally difficult. If you could do one, it would probably be easy to do the other.