 Solution

Take $K=\mathbb{Q}$. For every positive integer $n$, the real number $\sqrt[n]{2}$ lies in $\overline{\mathbb{Q}}$ and has degree $n$, by Eisenstein's criterion at $2$. If $\overline{\mathbb{Q}}/\mathbb{Q}$ had finite degree, all its subextension degrees would be bounded by that degree, contradicting this unbounded family. Therefore $[\overline{\mathbb{Q}}:\mathbb{Q}]$ is infinite.