Fair enough, I'm starting to read up on this. Obviously, you can define sqrt(2)^sqrt(2)^... to mean the (unique?) attracting fixed point of x -> sqrt(2)^x. But that makes your explanation a bit circular (it is so because it's defined to be so). Probably that's the only useful definition, but that fact is certainly not trivial or self-evident.
Well, the thing is that you can test whether a fixed point is an attracting fixed point; per Wikipedia (which I am relying on for the details of the test because I haven't used this for years, but the test looks correct), any fixed point x of a function f which has an open neighborhood where f is continuously differentiable and f'(x) < 1 is attracting.
