Let $\Gamma$ be a transcendental analytic curve in $\mathbb{R}^2$. I am interested in the topology of its rational points $\Gamma(\mathbb{Q}):=\Gamma\cap\mathbb{Q}^2$.

We know by Pila-Wilkie that if $\Gamma$ is compact then it has few rational points.

Is it true that $\Gamma(\mathbb{Q})$ is locally finite? Is it possible for $\Gamma(\mathbb{Q})$ to be dense in some open subset of $\Gamma$?

Any example/counterexample will be appreciated.