For any odd positive integer $k$, the map $f_k(x):=x^k$ is both an analytic function and a homeomorphism on $\mathbb{R}$, since $f_k^{-1}(x) = \sqrt[k]{x}$ is bijective an analytic. In fact, this shows that it is a biholomorphism.
Are there examples of biholomorphisms from $\mathbb{R}$ to itself, more generally examples of analytic homeomorphisms (not there is no requirement for the inverse to be more than continuous here), which are not polynomials?
Yes. Take $\sinh$, for instance.
On the other hand, $x\mapsto\sqrt[k]x$ isn't even differentiable at $0$. So, it's not analytic.