There is a unique formal power series solution with $f(0) = 0$ and $f'(0) = 1$. I had supposed that the coefficients would all be positive, which would imply that they are smaller than for $\exp(x)$ itself and thus that $f(x)$ is entire. No such luck. Maple gives me this:

$$f(x) = x + \frac{x^2}4 + \frac{x^3}{48} + \frac{x^5}{3840} - \frac{7x^6}{92160} + \frac{x^7}{645120} + \frac{53x^8}{3440640} + \cdots.$$

This doesn't say much about the possible radius of convergence of this series. On the other hand, expecting it to be entire may have been naive from the beginning, because it seems unlikely that $f(f(x))$ would be periodic in the imaginary direction.

Since Michael Lugo has found evidence that the Taylor series has zero radius of convergence, it's not a very good way to describe or even define $f(x)$. Is it clear that there is a unique $f(x)$ which is convex (at least for $x \ge 0$), and that that $f$ is smooth at 0 and real analytic away from $0$? There is a book on fractional iteration of functions that presumably addresses these issues.

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There is a unique formal power series solution with $f(0) = 0$ and $f'(0) = 1$. I had supposed that the coefficients would all be positive, which would imply that they are smaller than for $\exp(x)$ itself and thus that $f(x)$ is entire. No such luck. Maple gives me this:

$$f(x) = x + \frac{x^2}4 + \frac{x^3}{48} + \frac{x^5}{3840} - \frac{7x^6}{92160} + \frac{x^7}{645120} + \frac{53x^8}{3440640} + \cdots.$$

This doesn't say much about the possible radius of convergence of this series. On the other hand, expecting it to be entire may have been naive from the beginning, because it seems unlikely that $f(f(x))$ would be periodic in the imaginary direction.