Infinite Degree Algebraic Field Extensions

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?

Another simple example is the extension obtained by adjoining all roots of unity.

Since adjoining a primitive $n$-th root of unity gives you an extension of degree $\varphi(n)$ and $\varphi(n)=n-1$ when $n$ is prime, you get algebraic numbers of arbitrarily large degree when you adjoin all roots of unity.

$\mathbb Q[\sqrt 2, \sqrt 3, \sqrt 5, \cdots]$, obtained by adjoining the square root of the primes, is an example because if you use just $n$ primes, you get an extension of degree $2^n$.

The field of algebraic numbers is important, as is the field of real algebraic numbers. There are plenty of other examples of the same nature. The field of Euclidean constructible numbers is an extension field of the rationals, of infinite degree over the rationals, that comes up "naturally."