Fractional iteration of x^2+1 at infinity and fractional iteration of exp

[JmsNxn]

Exp(∞) may be a larger infinity.

Hmmmmm, you'd have to qualify that using some kind of framework. No idea what that would be. You could use something like Hardy spaces, and refer to \(1/\exp(\infty)\) in the right half plane as smaller than \(1/\infty\) in the right half plane. But then, you'd have to qualify how you mean this. Typically we're not referring to growth hierarchies. And they don't apply to Tommy's comment.

Exp() = beth 1.

\(\aleph_0 \neq \infty\)

Aleph zero is infinite.

Yes, but not in the sense of complex analysis limits. The point \(\infty\) on the Riemann sphere, is not the same thing as Aleph zero.