I really, truly never thought I'd be arguing about Naruto of all things on LW :-)
Let's enumerate Naruto's ninja skills (that not all ninjas have), the important ones that let him progress in the plot and make him stand out among other ninjas, in approximate order of acquisition:
Fast healing. Due to being the host of the Kyubi. Does not involve training.
Shadow clones. Gained miraculously in a few hours, where few others ever master it even after long study, because he was under emotional stress and Had To Win. Technique does not improve with practice, it's merely proportional to the amount of chakra he has or expends (which grows during the series). Over time he finds new ways of using it tactically, but he doesn't get better at using it.
Kyubi involuntary take-over rage mode (e.g. to defeat Orochimaru, and Pain). Obviously does not require training.
Rasengan. He trained for some time, but couldn't master it. Then he thought of a clever trick that let him master it and would have worked with no training at all. This is the only skill he improves over time, presumably through training and experience, creating bigger and fancier versions. Score one for Naruto. Various combinations with other techniques (shadow clones, nature chakra) also imply training offscreen.
Nature chakra (senjutsu). Only 4 ninjas other than him are named as ever mastering it, over the past 4 generations. He trained for a few months and became much better than at least one of them who practiced it all his life (Jiraya), if not the other three (Orochimaru, Minato and Hashirama), despite a handicap none of the others had (the Kyuubi).
Control of Kyubi's chakra. He trained for several weeks to accomplish it. We don't know if this unusual, because the only other to do so is Bee, who taught him, and no-one else ever got the chance to try that we know of.
Voluntary submission and aid of the Kyubi and his chakra, distributing it to others, assuming the Giant Kyubi Narutoform in battle, etc. Achieved making friends and influencing people - "just" being nice. To be fair, he does train a lot at being nice - at least he practices it all the time.
Various world-breaking powers granted by whatshisname, the spirit of his ancestor possessing Naruto.
I think the only thing that actually improved with use and training is the Rasengan, and even that didn't require any training to gain initially.
Now of course he also knows the basic techniques that all ninjas know, and he does train the acquire and to improve them. Water-walking, chakra nature shaping, even simple things like knife throwing. But he trains at them precisely because everyone does. All the things that make him powerful, important, interesting, and not-dead-long-ago are plot armor and unearned gifts. Take all of them away and he literally wouldn't have made it out of the Ninja Academy.
Compare that with the way e.g. Lee trains to get skills and to improve them.
I think the only thing that actually improved with use and training is the Rasengan, and even that didn't require any training to gain initially.
But
He trained for a few months and became much better than at least one of them who practiced it all his life (Jiraya)
He trained for several weeks to accomplish it. We don't know if this unusual,
To be fair, he does train a lot at being nice - at least he practices it all the time.
Now of course he also knows the basic techniques that all ninjas know, and he does train the acquire and to improve them.
You're ...
As people who care about rationality and winning, it's pretty important to care about training. Repeated practice is how humans acquire skills, and skills are what we use for winning.
Unfortunately, it's sometimes hard to get System 1 fully on board with the fact that repeated, difficult, sometimes tedious practice is how we become awesome. I find fiction to be one of the most useful ways of communicating things like this to my S1. It would be great to have a repository of fiction that shows characters practicing skills, mastering them, and becoming awesome, to help this really sink in.
However, in fiction the following tropes are a lot more common:
Example of exactly the wrong thing:
The Hunger Games - Katniss is explicitly up against the Pledges who have trained their whole lives for this one thing, but she has … something special that causes her to win. Also archery is her greatest skill, and she's already awesome at it from the beginning of the story and never spends time practicing.
Close-but-not-perfect examples of the right thing:
The Pillars of the Earth - Jack pretty explicitly has to travel around Europe to acquire the skills he needs to become great. Much of the practice is off-screen, but it's at least a pretty significant part of the journey.
The Honor Harrington series: the books depict Honor, as well as the people around her, rising through the ranks of the military and gradually levelling up, with emphasis on dedication to training, and that training is often depicted onscreen – but the skills she's training in herself and her subordinates aren't nearly as relevant as the "tactical genius" that she seems to have been born with.
I'd like to put out a request for fiction that has this quality. I'll also take examples of fiction that fails badly at this quality, to add to the list of examples, or of TVTropes keywords that would be useful to mine. Internet hivemind, help?