Define property $A_R$ for an analytic function $f(z)$ as
$1)$ $f(z)=0$ has exactly one solution being $z=0$ for $|z|<R$ where $R$ is a radius. And $f(z)$ is analytic within the radius $R$ (centered at $0$).
$2)$ $f(z)=z$ has exactly one solution in the complex plane $=>z=0$ (follows from $1)$ )
$3)$ $f(z)$ is not a polynomial.
Define property $B_R$ for an analytic function $f(z)$ as
$1)$ $f(z)=f_1(z)$ with property $A_R$.
$2)$ $f_i(z)= \ln(f_{i-1}(z)/z)$ for every positive integer $i$.
$3)$ $f_i(z)$ is entire and has property $A_R$ for every positive integer $i$.
I think there are functions $f(z)$ that have property $B_R$ for some $R>1$.
Is that true ?
Is there a solution for any $R$ ?
Is there a way to describe $f_i$ easily ? How does $f_i$ behave or grow with respect to $i$ and/or large $z$ ?
Not a complete answer but to much for a comment.
If $|f(z)| < |e^{e^{...^z}}|$ , in other words if $|f(z)|$ grows slower than a repeated exponential function then :
For sufficiently large $i$ and large $z$ the equation for the fixpoints starts to get like $ln(ln(...ln(z)...)= z$ hence we can conclude that the norm of the fixpoints is bounded then.
Tommy1729 suggested trying $f(z) =$ $z/a$ or $a* sinh(z)$ for $a>\frac{3}{2}$. ( without thinking about it )