In I. Martin Isaacs Algebra: A Graduate Course, Isaacs uses the field of algebraic numbers $\mathbb{A}=\{\alpha \in \mathbb{C} \; | \; \alpha \; \text{algebraic over} \; \mathbb{Q}\}$ as an example of an infinite degree algebraic field extension. I have done a cursory google search and thought about it for a little while, but I cannot come up with a less contrived example.
My question is
What are some other examples of infinite degree algebraic field extensions?