TITLE: An entire function all whose forward orbits are bounded
QUESTION [5 upvotes]: 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 $f(z)=\lambda z,\; |\lambda|\leq 1$.

REPLY [10 votes]: 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, the escaping set of a non-linear entire function is nonempty.  
The reference for this is On the iteration of entire functions by Eremenko.