Function in H(curl) $\cap$ H(div), but not in H1 it is well known, that for a non-convex domain $\Omega$ the space  $H^1(\Omega, \mathbb{R}²)$ is a proper subset of $H(curl) \cap H(div)$. 
Here, $H(curl) = \{v \in L²(\Omega)², \nabla\times v = \partial_1v_2-\partial_2v_1 \in L²(\Omega)\} $ and $H(div)=\{v \in L²(\Omega)², \text{div } v = \partial_1v_1+\partial_2v_2  \in L²(\Omega)\}$.

I am looking for an example, where $ v \in H(curl) \cap H(div)$, but not in $H^1(\Omega, \mathbb{R}²)$. Furthermore, I want to have the constraint div v=0. Lets consider $\Omega=(-0.5,0.5)²\backslash [0,0.5]²$. 

In Paper, there might be an example given by equation (5.2)
$v=\nabla \times (r^\frac{2}{3} \text{ cos}(\frac{2}{3} \theta-\frac{\pi}{3}))$. Is this one an example, I am looking for?

Thank you very much for your help.
The example you give from the paper is a little bit obscure to me. I propose another solution/perspective, which is quite general (but does not provide an explicit expression, at least in principle) for what concerns counter-examples in the context of $H(\text{curl}),H(\text{div})$. The main idea is to go back to the Laplace equation, which is very easy to deal with, and everything is known about it. 

Consider the (weak) solution $u \in H^1(\Omega)$ of the boundary value problem

$$
\begin{cases}
\Delta u = 0 \qquad \text{in  } \Omega \\
u = g \qquad \text{on } \ \partial \Omega,
\end{cases}
$$
where $g \in H^{1/2}(\partial \Omega)$. It is well-known that $u \notin H^2(\Omega)$ if the domain is not enough regular (non-convex, with non-smooth boundary...). 
Define now

$$ \mathbf{w} := \nabla u; $$
then $\mathbf{w} \in \mathbf{L}^2(\Omega) $ since $u \in H^1(\Omega)$, $\operatorname{curl} \mathbf{w} = \mathbf{0}$ since $\mathbf{w}$ is a gradient and $\operatorname{div} \mathbf{w} = \operatorname{div}(\nabla u)= \Delta u = 0$ by construction.

On the other hand, it can happen that $\mathbf{w} \notin (H^1(\Omega))^3$, because $u$ is not necessarily in $H^2(\Omega)$: see remark above.
The equation (5.2) from above is indeed the example, I was looking for. See for example the book of Peter Monk on Finite Element Methods for Maxwell's equation on page 76 for the explanation.