Hi c0mtesse,
Unfortunately, the 14 day trial course won't disappear after the 14 days of trial. It basically stays in the software, and what happens after 14 days (if you haven't licensed your LR) is that your software just gets deactivated.
At the moment, there isn't a way to remove the trial course, although the program should always have the course you last opened or used up on top - this means that if you haven't used your trial course in a long while, it should just keep on going to the bottom of the list. It shouldn't always select the trial course at all when you've never used it.
I'm interested in finding out what version of Little Reader you're running so I can test this out.
Kindly let me know what your version number is so we can check for you.
http://support.brillkids.com/entries/20426777-how-to-find-your-brillkids-software-version