Although the error occurs in Utils.LoadExtensions(), the main error message at the top is:
"Access to the path 'D:\hshome\gertbrigsted\blog.manisto.dk\App_Data\datastore\extensions\BBCode.xml' is denied."
And I'm prompted for username/password credentials before the error message appears, and then it appears. This means that the blog is trying to write to that file, but doesn't have permissions to write to it.
That's good you have a full trust server, but that doesn't necessary mean that Write permissions are setup on the App_Data folder. If your site is on a shared hosting plan, usually there is a control panel where you can control file permissions.
You would want to give full access to the App_Data folder for the ASPNET account (sometimes aka Network Service).
Other data also needs to be stored in App_Data, such as blog posts, comments, etc. So getting file permissions setup on App_Data is imperative.
If you're not sure, you can ask the hosting provider about setting up file permissions.