Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$. On the other hand, the algebraic completion of $\mathbb{Q}$ is $\overline{\mathbb{Q}}$, and if we complete it with respect to the Euclidean norm now, we get $\mathbb{C}$. So it indeed terminates and in fact the result is the same field/space.

Now, another example is the topological completion of $\mathbb{Q}$ with respect to the p-adic norm, giving us $\mathbb{Q}_p$ the p-adic numbers, then the algebraic completion $\overline{\mathbb{Q}_p}$, and then once more with topological completion under p-adic norm, we get $\mathbb{C}_p$ which is the end.

Maybe this question can be better phrased in terms of ascending chains/homology but if anyone has any insight, that would be great.

"On the other hand, the algebraic completion of $\mathbb{Q}$ is $\overline{\mathbb{Q}}$ and if we complete it with respect to the Euclidean norm now, we get $\mathbb{C}$"is a bit confusing, because for a finite extension of $\mathbb{Q}$, there is no unique extension of the norm (the Euclidean norm on $\mathbb{Q}[\sqrt{2}]$ is not invariant under the Galois action). $\endgroup$ – YCor Feb 8 at 18:058more comments