Rationalist fiction: Read Harry Potter and the Methods of Rationality, Time Braid and Luminosity. Time Braid may seem like a Mary Sue story, but the frequent need to overcome mind manipulations / conditioning makes it well worth reading if you like rationalism.
Some very well written, shorter stories: The Metropolitan Man and The Dangers of Foresight. Foresight can be read without reading the prequel, you'll just lack an explanation for how Anikan ended up there.
Stories I enjoy reading tend to follow these rules:
The (first) Three Laws of Fanfiction by Less Wrong
Rule Two: Originality isn't easy, but it is simple: Just don't do stuff that's already been done. Even if all of your other characters are going to be absolutely true to canon, you still shouldn't have Harry Potter facing the same three challenges in the Triwizard Tournament because we've already read about them a thousand times. Put in three different challenges. Seriously. It can't hurt. Don't just go through the same events everyone has read about a thousand times before. Writing fanfiction lets you borrow the characters and the world; it doesn't exempt you from needing to surprise the readers and give them something new to read.
Rule Three: The premise of a story is a conflict and its resolution - someone with a goal, which they take action to achieve, and severe obstacles that they must replan to deal with (not just speedbumps along the way), and some ultimate resolution of the conflict in which the people and their situation have changed. "What happens if the Terminator is sent back in time to kill Voldemort" is not a story premise, just a fleeting mental image. "What happens if Harry Potter is under constant attack by shape-changing robot assassins" is still not enough of a premise. "Harry Potter is under constant attack by shape-changing assassins, and by the time he manages a spell to wall off the future he's already learned not to trust anyone" could maybe be a story's premise (though you wouldn't put that in the summary, or tell any reader that until the story had ended). You can change this plan later - but you should at least have one to start with.
So if you have a lovely mental image of Frodo with a lightsaber:
1. Figure out how to make his life more difficult, to make up for the lightsaber.
2. Decide what's going to happen differently in your fanfiction than in the other ones you've read.
3. Know what Frodo wants and what's going to get in his way, and have a plan for how it will all end.
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