**Edit:** I revise the question according to the comment of Gabe Conant.


What is an example of a non constant entire function $f:\mathbb{C}\to \mathbb{C}$ which satisfy the following?:



> 
> For every $z\in \mathbb{C}$, the sequence $z,f(z),f^2(z),\ldots,f^n(z),\ldots$ is a bounded sequence but $f$ is not in the form $f(z)=\lambda z,\; |\lambda|\leq 1$.
> 
> 
> 



An entire function all whose forward orbits are bounded
Given an entire function $f\colon\mathbb{C}\to\mathbb{C}$, the *escaping set*, $I(f)$, is the set of $z\in\mathbb{C}$ such that $f^n(z)\to\infty$. Per the [Wikipedia article](https://en.wikipedia.org/wiki/Escaping_set#Properties), the escaping set of a non-linear entire function is nonempty. 


The reference for this is *[On the iteration of entire functions](http://www.math.purdue.edu/~eremenko/dvi/banach.pdf)* by Eremenko. 


