Looking for examples where $f(z)=\operatorname{inv} \int_{0}^{z} g(z)\, dz$ with $f(z)$ entire and $g(z)$ not meromorphic.

Question

I'm looking for examples where $f(z)=\operatorname{inv}\int_{0}^{z} g(z) \, dz$ with $f(z)$ entire and $g(z)$ not meromorphic. For clarity, by $\operatorname{inv}$, I mean the functional inverse. Additional conditions are:

1. $g$ must have a positive radius of convergence around $0$.

2. $f$ is not a polynomial.

3. $f$ is not $\sin(az+b)+c$, $\cos(az+b)+c$, $\tan(az+b)+c$ or their hyperbolic analogues.

Notice that $f(0)=g(0)=0$ must hold.

I know the Lagrange inversion theorem and I know how to express the derivatives in terms of contour integrals. (http://en.wikipedia.org/wiki/Lagrange_inversion) (http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula)

But I'm still stuck. Maybe abelian functions are what I want?

I'm still confused about how things work. I know of course that bounding (all) the derivatives could prove a function to be entire—which is not always easy either?—but that knowledge did not help me so far.