Given a ordered field $K$ such that every positive element $0<x$ has a root in $K$ we can show that any endomorphism $f:K\rightarrow K$ preserves the ordering of $K$ and turns out to be $\text{id}_K$. It follows that $\text{Gal}(K/\mathbb{Q})=\{\text{id}_K\}$.
So now I want to find some examples of such $K$, we of course have $K=\mathbb{R}$. Another example I have found is $K=\mathbb{A}\cap\mathbb{R}$ where $\mathbb{A}\subset\mathbb{C}$ is the algebraic closure of $\mathbb{Q}$. Do you know any other examples (or families of such examples for that matter)?
Regards
A large class of reasonably well-studied examples is given by the real closed fields, which can be defined as the ordered fields such that every positive element has a square root and such that every polynomial of odd degree has a root. The Wikipedia article gives several examples, such as the nonstandard reals.
By the way, I would not personally refer to the automorphism group of a non-Galois extension as a Galois group; such extensions e.g. don't have a Galois correspondence. They're just automorphism groups.