Suppose $L/K$ is a field extension and that there is a uniform natural number $n$ such that for all $\alpha \in L$ the degree of $\alpha$ over $K$ is at most $n$. Is the field extension $L/K$ necessarily finite?
If not, can you construct a counterexample.
Let $\{X_n\}_{n=1}^{\infty}$ be a set of indeterminates, and set $K=\mathbb{F}_2(X_n^2:n\in\mathbb{N})$, $L=\mathbb{F}_2(X_n:n\in\mathbb{N})$.
The set $\{X_n\}_{n=1}^{\infty}$ is a $K$-basis for $L$, so $L/K$ is not a finite extension. But if $f\in L$ then $f^2\in K$ (because the cross terms vanish), so $f$ has degree at most two over $K$.