What makes a Mary Sue is not so much your character's past as it is how you write that character. Aka, a character could be pretty and nearly everyone could be in love with her but as long as that does not have other characters acting like they would never dream of to win her love (like say, Ash jumps off a cliff in grief, or kills Pikachu), it could make an okay character. Basically, if the world changes for the character (say they are in space and she just happens to know how to work a rocket for not reason without having any training) rather the character being changed by the world (a spoiled rich girl has to learn to cook on a camp fire to be a pokemon trainer, the shoes she never goes without are awful for walking give her blisters etc)
Basically, a Mary sue is a character who is never wrong, and the only things that hurt her are things that aren’t her fault. A good character's growth comes from mistakes that they made and have no one else to blame for but themself (Of course, they can blame others but it would not be true
. Everyone loves to blame someone else).
Example:
A character gets kidnapped by..Team Rocket. (These are just samples, I don't really write like this)
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Mary Sue was just sleeping in her tent when The Big Evil Rocket Member, seeing how pretty she was, decided to steal her away! A moment later her friends woke up, no longer feeling Mary Sues golden glow, Oh no, who will save Mary Sue?
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Alice had gotten up in the middle of the night from her friend's tent...If you could call them friends, she was sure at least half of them only looked after her because her parents made them promise to.
Her reasons for not sleeping were perfectly clear: All she cared about was getting the best pokemon and the best of the best only came out at night here.
She was chasing blindly after the HootHoot in a power hungry rage moments later (never mind that she forgot to bring any pokeballs), when she lost her footing in the grass and triped.
"Well, well, what do with have here?" Remarked the Rocket sleepily.
"A new trainer your going to let go like this never happened?" Peeped Alice
"I don't think so." He replied
The next morning the other trainers only took a moment to call Alice's parents and journey on....
*lol* Bad I know, but I hope that helps just a little.
*suddenly feels like writing a much better verison of the Alice character story*