Or, obviously, Galadriel's secret love-child (Aragorn's unacknowledged daughter) who runs off to join the Company of the Ring, sorts out Boromir's problems, out-magics Gandalf, out-fights Aragorn during the melodramatic scene in which she reveals her true identity, demonstrates herself to be so spiritually elevated that the Ring has no effect on her, and wins Legolas' heart forever.
As you can expect, Mary Sue stories (most of which are probably written by people in the lower reaches of the Geek Hierarchy) are ripe targets for mockery. There is a LiveJournal community dedicated to examining the most egregious examples of Lord of the Rings/Harry Potter fanfic and ripping them gleefully to shreds here; some of the examples (like "Kairi", the raven-haired elf whose abusive father is in league with Sauron, or, indeed, this, or this) sound thoroughly cringeworthy. Anyway, go and read it; it's a laugh-riot.
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