I've discussed the pros and cons about static blogging many times before. In summary, the cons are that, well that your blog is static. No surprise there.
Fortunately, there are ways to push the limits of static blogging so that it is not quite so static.
Unless you are self-hosting your Octopress or Jekyll blog, you are using another site to host your blog. I host mine on Github. They have a web server running 24x7 to host it. But that's not my problem. I just use their (free) services.
I understand that website hosting isn't a terribly exciting example of stretching the bubble of static blogging, but it is an example nonetheless. Dynamic blogs require specialized software running on their hosting computer to work. Static blogs tend to be able to run on generic web servers but there still must be a web server or there's no blog.
Just to be clear, by dynamic data I mean that the data displayed on the page may change without a republication of the blog. I am not necessarily referring to a scrolling widget or something. Just dynamic data.