Repeated Differentiation Leading to Tetrationally Fast Growth

[Catullus]

No, Daniel's right, there is no such function--at least no such analytic function. I doubt there's even a smooth function.

Every analytic function must satisfy:

\[
\lim_{n\to\infty} \left(\frac{f^{(n)}(k)}{n!}\right)^{1/n} \le 1
\]

And it's safe to say:

\[
\lim_{n\to\infty} \left(\frac{^n a}{n!}\right)^{1/n} = \infty\\
\]

 How many times real differentiable do you think it could be?
Do you have any more guesses about that?