I noticed something on this page, that may just be coincidental:
http://www.lmfdb.org/LocalNumberField/
From inspecting the table there, you can conclude that most of the interesting extensions of the p-adic fields take place in the degree $p^n$ extensions. And what's more interesting than that is that they seem to lie in a recursive sequence. Look at the numbers 10, 26, 50, and 122. They fit in the following broader recursive sequence:
$2 + 8\cdot1 = 10$
$10 + 8\cdot2 = 26 $
$26 + 8\cdot3 = 50$
$50 + 8\cdot4 = 72$
$74 + 8\cdot5 = 122$
If I had to guess, I would say that the number of degree 13 extensions of the 13-adic field would be:
$122 + 8\cdot6 = 170$
Edit* I just realized that this recursive formula is given more simply as $p^2 + 1$
For example here, http://arxiv.org/pdf/1105.5520.pdf, you can see table of all totally ramified extensions of degree $p$ (plus 1 unramified extension). We have $(p-1)^2-1+p+p+1=p^2+1$.