Sean - write the Shawn and Feeny ones!
1. Cory Matthews
It really surprised me how many people seemed to think anyone else would be Number 1. Yes, Sean's favorite character is Shawn, and yes mine's Eric, but this is clearly Cory's franchise. We found it difficult whittling down Shawn and Eric to a handful of Signature Episodes, with Cory it's literally impossible. Shawn and Eric almost always have fun stuff to do, but the episodes that truly examine them, where they're the ones driving the story... you're probably looking at 20 or so for Shawn, and maybe 10-15 for Eric. And the remaining 100+? Basically all Cory. We take him for granted, and he's few people's favorite character, but Cory's the wheel that keeps this all turning. All the characters have distinct, important relationships with him (well, besides, like... Rachel.) and most of them exist solely as different kinds of foils to Cory. Feeny's old and wise to show that Cory is young and naive, Shawn is edgy and abandoned to show that Cory is safe and beloved, Eric is charming and fun to show that Cory is geeky and awkward. It's all about Cory.
If that's not enough, he also a main character on an entirely different shows. Seriously, even if I could understand someone not thinking Cory would beat Shawn, Eric, and Feeny just ranking BMW characters, how could any of them beat him when you combine them both? Cory's an important part of Girl Meets World - combining the roles of Feeny and Alan in the story (while retaining his own unique personality) to really become a co-lead along with Riley and Maya. Meanwhile, the other three appeared in only like a handful of episodes each. How could they beat Cory? They couldn't.
Also, Cory's a great character. I'm going to hustle past this part, but he's really a lot of fun. If you boil them down to their main role, a lot of shows have characters like Shawn, Feeny, and even Eric. Few shows have someone exactly like Cory. People like Cory are usually fun supporting characters, not male leads. To go through such a traditional boy turning into a man story with such a flawed, neurotic, emotional little weirdo was a real treat.
2. Shawn Hunter
When I think of Shawn, I like to think of episodes like The Eskimo, B&B, Graduation, Turkey Day, The Heart is a Lonely Hunter, right, where we've got this guy who, most of the time, genuinely believes he's worthless trailer trash, but occasionally finds the strength in himself to rise above it, or is finally willing to lean on his friends enough to have them raise him above it. That's the Shawn I love. Especially, regarding the latter part of that, it's not so much that he lets Cory raise him up, but more that he trusts his best friend so much that he's willing to believe in the version of himself that Cory sees. And that's why City Slackers is so great, and why his relationship with Turner is so great, because he also learns to trust Feeny and Turner (and Alan, in those rare magical moments) enough to believe them when they say he can rise above his station. It doesn't matter when Chet tries to give Shawn a pep talk, because Shawn knows better than anyone that Chet's words aren't worth shit. So yeah, it's really the trust Shawn builds for his non-blood family and subsequently trusting their faith in him, that's where the money is. And then it's that exact dynamic that gets spun around when he gets to know Maya. It's that exact thing, where she learns to trust Shawn enough to believe in his vision of her. It's just god damn fantastic, it really is. And there are all these other themes, right, further exploration of family, charity and his pride, poverty, womanizing, I'm just realizing this now but did Shawn get the full course of the deadly sins over the series? I don't think I can place gluttony, but the other six definitely gave Shawn at least one awesome episode each. But I digress.
The problem is, I have this little demon living on my shoulder about Shawn. It's disingenuous to talk about his amazing stories without admitting that The Fugitive happened. Cult Fiction happened, Ode to Holden Caulfield happened, Family Trees happened where he was an alcoholic for all of 20 minutes, his mostly uninspired arc with Angela happened, and his relationship with Jack was never worth more than their scenes with Chet. He worked for the mob for an episode! There is some seriously ridiculous, whiny crap mixed in with the magic, and it's frustrating, especially because I think that's what Rider Strong remembers the most. And I can't really blame him either. It's probably a lot harder to forget Cult Fiction when you actually had to act it out.
I guess the core question is this. Does the fact that Shawn is an unrealistically tortured mess automatically neuter his development arcs? Sure he complains all the time, but incredibly awful shit is constantly happening to him. So maybe it depends how far your suspension of disbelief can go. I honestly don't know. Maybe I just have to watch the series a few more times.
There was some debate about Shawn's placement in the ranking early on, but Girl Meets World pretty much dissolves any chance of debating this placement. After his growth as an adult, his continued relationship with Cory, his new relationships with Maya and (in particular) Katy, on top of everything from Boy Meets World, he is unambiguously in the top two.
3. George Feeny
Copying from a comment I wrote earlier, for Feeny I wanted to talk about his running theme of love, in particular when he lectures Cory in the pilot, "You SHREWD little observer of the human condition, having lived SO little, have already figured out the greatest mystery in the universe?" Or something like that, that's from memory. And in City Slackers his romantic diary helps Shawn finally relate to him, Eric consistently relates to him in romantic matters, it goes on and on. It's not his most important character trait, but it doesn't get the recognition it deserves so I wanted to shine some light on it. This short paragraph is obviously still not the recognition it deserves, but hopefully you'll keep an eye open for it the next time you watch the show.
We all know why Feeny is great. He's a legend. People who barely watch this show know who he is and they love him. More importantly, they love what he represents. And if absolutely nothing else, this show managed to convince people of the importance of great teachers. Girl Meets World tried to do that and mostly failed, so let's not make the mistake of thinking it's a minor accomplishment.
I keep thinking that I should start this epic description of Feeny being this teacher who's stern but fair but wise but kind and so on, but you guys know all that already. This isn't baby's first Feeny discussion, and the real heart of the truth is that he means something special to all of us. I'm not going to tell you what makes him special because you already know. Even if it's different from what makes him special to me.
As far as the rankings go, like Christian says in the Eric section, his influence reaches far and wide. To everyone. Except Amy. He is a relentless force for good and wisdom for seven straight seasons, and that kind of consistency should honestly be impossible. Ultimately, he's behind Shawn and Cory because he's never really the focus. The show isn't about Feeny. He does still meet the world on occasion, thanks to the other fellows on this list, but not enough to be placed above Cory or Shawn.
4. Eric Matthews
With the possible exception of Will Friedle, there is no one who likes Eric Matthews more than I do. But Eric had to be fourth. Is he the funniest character on this, a comedy show? Absolutely. Without question. But while he's always funny, and usually hilarious, the facts are that the truly great Eric existed in only two seasons: Season 4 and 5. Season 1 is basically a different guy, Season 2 and 3 feel like beta-versions, Season 6 started to go downhill, and Season 7 hit the bottom of the hill and crashed and burned. The Eric we got in GMW was sort of a grab-bag of the different kinds of Eric, but ultimately more latter days Eric than anything else.
Shawn wasn't always consistently written either, but he usually was, and he did feel like he had one long arc - even if that arc involved him having to re-learn lessons a lot, and even if that arc involved him being awfully annoying at times. Eric's Season 3 to 4 arc is, in my opinion, better told than Shawn's was on his best day. But it lacked the consistency, the longevity, the overall impact, and the legacy of Shawn's arc.
Plus day-to-day, Shawn was usually just more important. The strength of Eric is how he could be used in any situation and bounce off any character and be fun. So, they often have him bounce off people in B stories, because usually Cory and Shawn are leading the A story. So he just feels less focused on. It's mainly a show about Cory going to school, and half the time Cory and Eric weren't in the same school. That's gonna mean less Eric.
Eric and Feeny was closer, but ultimately the show's focus on the importance of teachers and mentors (embodied by Feeny) the way he ultimately has become kind of a pop culture teacher icon, and his lasting legacy of the same on GMW pushed him over the edge. Plus, really, I think Feeny winds up with more significant well-developed relationships on BMW than Eric does. Eric barely interacts with half the cast. Feeny interacts with everyone.