Tower of Galois extensions and completions Consider the following infinite tower of Galois extensions
$$
\mathbb Q\subsetneq L_1\subsetneq\ldots\subsetneq L_n\subsetneq\ldots
$$
and for each $L_n$ fix an absolute value $w_n$ such that: ${w_n}_{|L_{n-1}}=w_{n-1}$ and $w_0=\vert\,.\vert_p$.
Now let $L_{w_n}$ be the completion of $L_n$ with respect to $w_n$. So we get the following tower:
$$
\mathbb Q_p\subset L_{w_1}\subset\ldots\subset L_{w_n}\subset\ldots 
$$

Does such tower stabilize, or the degree of the fields (over $\mathbb Q_p$) goes to infinity like in the non-completed case?

In general we have $[L_n:\mathbb Q]\ge [L_{w_n}:\mathbb Q_p]$
 A: It very much depends on the extension.
For an example where it does not stabilise, just take
$$\mathbb Q\subset\mathbb Q(\zeta_p)\subset\mathbb Q(\zeta_{p^2})\subset \mathbb Q(\zeta_{p^3})\subset\cdots.$$
Since $p$ is totally ramified in $\mathbb Q(\zeta_{p^n})$, we have $[\mathbb Q(\zeta_{p^n}) :\mathbb Q]=[\mathbb Q_p(\zeta_{p^n}) :\mathbb Q_p]$ for all $n$.
For an example where it stabilises, let $a_1, a_2, \ldots$ be the squarefree integers such that the Legendre symbol $\left(\frac {a_i}p\right)=1$ (assume $p \ne 2$, but it's not too hard to construct examples with $p=2$ along similar lines). In particular, the set $\{a_i\}$ is the (infinite) set of squarefree integers in obeying certain congruence conditions modulo $p$.
Then
$$\mathbb Q\subset\mathbb Q(\sqrt{a_1})\subset \mathbb Q(\sqrt{a_1}, \sqrt{a_2})\subset\mathbb Q(\sqrt{a_1}, \sqrt{a_2},\sqrt{a_3})\subset\cdots$$
is an infinite tower. But by construction, $p$ splits completely in every $L_n$. In particular, $L_{w_n} = \mathbb Q_p$ for all $n$.
