If $K\_n$ is the field $\mathbb{Q}\_p(\mu\_{p^n})$, then it's easy to see that the relative different $\mathcal{D}(K\_n / K\_{n-1})$ is $(p)$ for all $n \ge 2$. 


What happens if I take an arbitrary, probably totally ramified, finite extension $L/\mathbb{Q}\_p$ and look at the tower $L\_n = LK\_n$? It's clear that $\mathcal{D}(L\_n / L\_{n-1})$ divides $(p)$, and one can show (using a general result of Tate on $\mathbb{Z}\_p$-extensions) that its valuation tends to 1 as $n \to \infty$; but is it true that it's equal to $(p)$ for all sufficiently large $n$?



What is the different in the cyclotomic tower over a finite ramified extension of Qp?
Take $K=Q\_p$ and $L/K$ finite. It is known that the sequence $\{ p^n v\_p( \mathcal{D}(L\_n/K\_n) ) \}\_n$ is eventually constant (it's basically the valuation of the different of the extension $E\_L/E\_K$ which you get from the theory of the field of norms). This and the transitivity of the different should allow you to answer your question.


As for the proof of my claim, see proposition 4.5 of <http://www.math.jussieu.fr/~colmez/monodromie.pdf>


