Here's how to find one: Let $E/F$ be a Galois extension with Galois group $G$, let $H$ be a maximal subgroup of $G$ whose index is not prime, and let $K$ be the fixed field of $H$.
The smallest group I know with a maximal subgroup of non-prime order is $S_4$, which has maximal $S_3$ subgroups of index $4$. This gives us the following example:
Let $E$ be the field $\mathbb{Q}(x_1,x_2,x_3,x_4)$ of rational functions in four variables with rational coefficients. The group $S_4$ acts on $E$ by permutation of variables. Let $F$ be the fixed field of this action, i.e. the symmetric rational functions. Let $S_3$ denote the subgroup of $S_4$ consisting of permutations that fix $x_4$, and let $K$ be the fixed field of $S_3$ (i.e. rational functions that are symmetric between $x_1$, $x_2$, and $x_3$). Then $K/F$ is a degree four extension with no proper intermediate extensions.