I don't understand why your step 3 is relevant. Observe that using r=sqrt(2) we have that the sequence r, r^r, r^r^r, r^r^r^r, ... does converge to 2. We are

I don't understand why your step 3 is relevant. Observe that using r=sqrt(2) we have that the sequence r, r^r, r^r^r, r^r^r^r, ... does converge to 2. We are not performing arbitrarily large exponentiation of x. Or, to pull a Calc 1 term, x^x^x^... does converge for x=sqrt(2).