I have used categories in so many different ways as I learn WordPress, it has gotten thoroughly confused at various times. I tried doing strange things to generate urls that worked for my purposes only to have the url double itself.
I did find that putting a single period in the “Category Base” did eliminate the term “category” in my url structure, but i also caused a weird instance where the archives page for child categories did not work for me.
I finally realized the term that works the best for me to place in the “Category Base”. I now use “archives” as the base instead of the default “category”. Now my urls will have the slug archives just before the remaining slug for the categories. When you click an item the url slug removes the archives portion and you have the SEO appropriate url slug.
Example. Say I have a parent category of news. The url would be http://myhomepage.com/archives/news/ and all my articles under news would appear. Then if I had a child category of local, my url would be http://myhomepage.com/archives/news/local/ and would list all my posts about local news. Then when you click on an article under local news the the url would look like http://myhomepage.com/news/local/article-01
This makes sense for me, as yes I want the clean url structure for the article, but when you are getting a list of articles for a category then you might as well have something to distinguish that this is a listing of categories. By default, WordPress will use /category/ as the base. However, I like the term /archives/ much better.