I'm not sure if this is the kind of thing you want...

Let $\wp$ denote the Weierstrass $\wp$-function with respect to a lattice. Then for any integer $n$, there is a rational function $f(x)\in\mathbf{C}(x)$ (depending on the lattice) such that for all $z\in\mathbf{C}$, we have $\wp(nz)=f(\wp(z))$. This is just a disguised version of the fact that for a point $P$ on an elliptic curve in Weierstrass form, the $x$-coordinate of $nP$ depends only on the $x$-coordinate of $P$. In the "complex multiplication" case, where the ring of endomorphisms of the lattice is bigger than $\mathbf{Z}$ (in which case it's a rank two subring of $\mathbf{C}$), then such $f$'s exist for any endomorphism $n$.

I think Ritt classified the entire functions that admit algebraic functional equations. (Maybe they're all algebraic functions of exponentials and $\wp$-functions?)

I'm not sure if this is the kind of thing you want...

Let $\wp$ denote the Weierstrass $\wp$-function with respect to a lattice. Then for any integer $n$, there is a rational function $f(x)\in\mathbf{C}(x)$ (depending on the lattice) such that for all $z\in\mathbf{C}$, we have $\wp(nz)=f(\wp(z))$. This is just a disguised version of the fact that for a point $P$ on an elliptic curve in Weierstrass form, the $x$-coordinate of $nP$ depends only on the $x$-coordinate of $P$. In the "complex multiplication" case, where the ring of endomorphisms of the lattice is bigger than $\mathbf{Z}$ (in which case it's a rank two subring of $\mathbf{C}$), then such $f$'s exist for any endomorphism $n$.

I think Ritt classified the entire functions that admit algebraic functional equations.

[Added: Although it's now clear that this is not the direction the OP wanted to go in, I thought I might add a bit more detail for the record.

The paper of Ritt's I was thinking of is "Periodic functions with a multiplication theorem", Trans. Amer. Math. Soc. 23 (1922), no. 1, 16–25. He restricts himself to periodic functions $g$, but apparently he does allow them to be meromorphic. What he proves is that if a periodic meromorphic function $g$ has a functional equation of the form $g(nz)=f(g(z))$ for some $n\in\mathbf{C}$ and rational function $f$, then '$|n|\geq 1'$. If $|n|>1$, then $g$ is one of the following: (i) a linear function of a function of the form $\cos(az+b)$, (ii) a linear function of a function of the form $\exp(az)$, (iii) a function of the form $\wp(z+a)$, $\wp'(z+a)$, $\wp''(z+a)$, or $\wp'''(z+a)$. If $|n|=1$, then $g$ must be one of a short list of rational expressions in exponential functions and derivatives of $\wp$-functions.

(NB I haven't thought about the argument. I'm just copying from his paper. He also gives more detail about which possibilities occur when.)

Interestingly, he also mentions a result of Poincare that for any rational $f$ satisfying $f(0)=0$ and $|f(0)|>1$, there is a meromorphic function $g$ with functional equation $g(f'(0)z)=f(g(z))$.

And last, from what I remember, the theory of Drinfeld modules gives examples of analytic functions (such as the Carlitz exponential) over local fields of nonzero characteristic with similar functional equations. It would be interesting to prove a Ritt-like converse result in that case.]