degree of extension of rationals adjoin infinitely many roots

Question

Consider a radical extension of $\mathbb{Q}$ in which infinitely many numbers are adjoined. Each of these numbers is the $n^{th}$ root of some integer, where $n$ can vary. Can any information be obtained about the degree of this extension, or its divisibility properties depending on what roots are adjoined?

Answer 1

The degree of the extension can be any positive integer, as well as infinite. @Timbuc's comment gives examples for $1$ and $\infty$. To get $n$ as the degree, adjoin the $n$th root of $2$ at the first stage, and $1$ at every other stage.