I am looking for an example of a field extension $k \subset F$ and a unital ring $R$ that is not a field such that
$$k \subset R \subset F.$$
I know if $F$ is algebraic over $k$, then this is not possible.
I am interested if it is possible with $k = \mathbf{Q}$ and $F = \mathbf{C}$.
There's plenty of rings with those properties, because $\mathbb{C}$ has transcendency degree $2^{\aleph_0}$ over $\mathbb{Q}$, so if you take any non empty subset $S$ of a transcendency basis, the ring $\mathbb{Q}[S]$ is not a field and is properly contained in $\mathbb{C}$; different subsets define different rings. So there are at least $2^{2^{\aleph_0}}$ rings like you require.
Not more, because that's the cardinality of the power set of $\mathbb{C}$, but of course those rings are not the whole set.
If you want one example, take any transcendental complex number $t$ and form the ring $\mathbb{Q}[t]$.