Is $\mathbb{R}$ an algebraic extension of some proper subfield?
Just to have at least one actual answer to the question, I'll convert the comment by PVAL to an answer.
Using Zorn's lemma there exists a transcendence basis $B$ of $\Bbb R$ over $\Bbb Q$: a maximal algebraically independent (over $\Bbb Q$) set of real numbers. Then $\Bbb Q(B)$ is a subfield of $\Bbb R$, and by maximality every element of $\Bbb R$ is algebraic over $\Bbb Q(B)$. So $\Bbb R/\Bbb Q(B)$ is an algebraic extension.