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Golbez, despite ultimately being revealed to be The Dragon for Zemus, maintains a sizable following due to being a cool Darth Vader Clone. His voice actor in the DS remake also helps.
Final Fantasy V has Gilgamesh, who is a full-on Breakout Character. Originally just The Dragon to Exdeath and a bumbling, cowardlyhammy one at that, he's made cameo appearances in almost every main game in the series, being worked into remakes of previous games and popping up from nowhere in the ones made after. He's so popular that Type-0 announced him to be in the game before it even came out, or before the playable cast was even fully revealed. Adding to his popularity is that where Gilgamesh goes, "Battle on the Big Bridge" is almost certain to follow.
Gilgamesh's boss, Exdeath, was nowhere near as popular, being a generic Tin Tyrant who wanted to destroy the world, mocked the heroes and threw out an Evil Laugh now and then. Then, Dissidia: Final Fantasy was released. Exdeath's popularity skyrocketed due to his hilariously hammy lines, his obsession with the VOID, and the sheer fun of being opponents up with a tree. He's also become a Fountain of Memes, with almost everything he says being quotable. This began before the game even had its English release when people noticed that one of his lines in Japanese sounded like "tasty arrow!", and could be chained to another certain attack for "tasty arrow...is yummy!" And then we got the English release..."Get the fruit!"
General Leo, so much so that there were rumors about ways to revive him.
In Final Fantasy VII Zack was very popular, so much so that despite having about a total of ten minutes of screentime in the original game, much of which was optional scenes, he got his own spin off Crisis Core. There are actually those who proudly declare "I loved Zack beforeCrisis Core!"
From the original game, Vincent Valentine, a secret character who's a Woobie, Destroyer of Worlds bishie with an angsty past and a cool gun. Even though recruiting him is optional, he's probably the most popular and well-known of the protagonists behind Cloud and Tifa. He got his own damn game Dirge of Cerberus, even though it wasn't any good.
Rufus, despite being reduced to a Big Bad Wannabe by Sephiroth's presence, manages to prove popular enough to return in Advent Children despite his apparent death.
In Final Fantasy VIII, Laguna, Ward, and Kiros, all proved extremely popular thanks to the interesting story in their segments, interactions that were both funny and played out more like real people, so much so that a portion of the fanbase like their segments better than the main story.
In Final Fantasy IX, Lady of War Beatrix and Black Waltz 3 are held in pretty high regard as well. While Beatrix serves as The Dragon to the Queen of Alexandria (thus remaining a prominent character throughout the game) and is a Magic Knight that absolutely puts Steiner to shame fifty times over, Black Waltz 3 was a bit player at best. His appearance most likely helped along with getting the most characterization of the Black Waltzes.
While Vivi is one of the main characters of the game, he also became one of the most popular characters in the series.
Freya was very clearly never meant to be popular or even widely liked, mostly being shoved to the sidelines after Cleyra's destruction on disc 2 and her whole species basically being a plot device. Despite this, she became one of the more popular characters.
Final Fantasy XAURON, even by people how aren't fans of the game like him due to his stoicism and showing himself to outright Badass every seen he appears in. It makes it pretty easy to forget that he's mostly around as The Mentor for Tidus and doesn't play too large a role in the story. It helps that in the game's largely poor English voice cast, his VA gives on the few good performances. He was so popular that Game Informer even put him in their list of the top thirty video game characters who defined the 2000s.
A very odd example: Shantotto of Final Fantasy XI is an absurdly popular Tarutaru who is known for being quirky, haughty, Crazy Awesome and possibly a megalomaniac (which is explored in the mini-Expansion Pack "A Shantotto Ascension"). It's amazing when you consider that her initial role in the game was as a quest starter for one insignificant quest chain and another quest chain that was restricted to Black Mages. She's actually also popular with the developers of Dissidia: Final Fantasy and, well...
Stepping off the main series, Final Fantasy Tactics deserves a few mentions. "Thunder God" Cid Orlandeau is regarded as one of the best warriors in all of Ivalice, everyone speaks of him in-story as a total badass. What happens when you finally recruit him? He surprisingly lives up to the hype. He's a total Game Breaker having access to all of the special sword skills of the other story characters use as well as being armed with one of the most powerful swords in the game. In the fanbase he's become the Chuck Norris of Ivalice.
Alica and Lavian, two generic Knight units that can join you at the beginning of Chapter 2 under the command Holy Knight Agrias Oaks. Aside from their set names they have nothing unique to them compared to and other female knight units and no lines of dialog. Despite this they are accepted as Canon members of the story by fans, often showing up prominently in fanfiction and fanart. Even the developer's seemed to think so as come time for the PSP remake the two were required for a late game romantic subplot as Mustadio tried to find out their former commanding officer's birthday.
When Balthier of Final Fantasy XII proclaimed himself as the "leading man", no one in the fanbase was quick to correct him. Also, there's Al-Cid Margrace, Larsa Solidor, and Judge Magister Gabranth. The former became a hidden playable character in Tactics A2, the latter was popular enough to make it into the aforementioned Dissidia. On the other hand, Montblanc (originally from Final Fantasy Tactics Advance) cements himself as a Breakout Character in this game, as he's probably the second most iconic element of the Ivalice games behind the Judges.
Even when Final Fantasy XIII wasn't out yet, Sazh Katroy found himself having a Darkhorse status from get-go. Not only his design is really different than other Nomura characters (Funny Afro non-bishonen, anyone?), he's got Badass Longcoat, dual-wields gun, has a chocobo on his afro and is a good guy extraordinary, so much that he is one of the fandom celebration triggers of the game. Certainly, Jihl Nabaat picked the wrong person to interrogate and boast on, so much that the moment she did that only in trailers (we don't know if she did that to Sazh because it's just duty or it's completely For the Evulz), she's immediately pegged as someone who crossed the Moral Event Horizon (you don't mess/fuck up with the Ensemble Darkhorse, bitch). Nomura likes him so much that said he would see Sazh in a Dissidia sequel over Lightning. When Lightning got announced for the sequel, fan speculation ran rampant over Sazh getting in as a secret character. He didn't, though.
In Final Fantasy XIV, Hildibrand Helidor Maximillian Manderville was originally conceived as a minor comic relief character, being a Small Name, Big EgoGentleman Detective character that completely contrasts against the rather grim main background story. However, he was so popular in the game's first incarnation that he was "resurrected" for the A Realm Reborn version of the game and given his own dedicated quest series.
When it comes to the villains of the Final Fantasy series, even previously-mentioned favorites like Exdeath and the Emperor have nowhere near the popularity of Final Fantasy VIs Kefka Palazzo and Final Fantasy VIIs Sephiroth. They are the most iconic baddies in the entire franchise. Kefka for being a creepy Monster Clown with hilarious lines and Sephiroth for being an intimidating Bishōnen. Like Gilgamesh, both characters owe some of their popularity to the music accompanying them: "Dancing Mad" from VI and "One-Winged Angel" from VII routinely appear on lists of the greatest Final Boss music ever, so popular they inspired tropes here on this very wiki - the former being the inspiration behind the name of the Laughing Mad trope, the latter as the Trope Namer for a narrative device that predates the game it was first named in.
This trope isn't limited to characters, some monsters and summons have gained a sizable following as well.
Tonberries are generally well-loved for looking adorable for such little killers, not to mention often giving decent rewards for (managing to) defeat them.
Chocobos and Moogles, but generally Chocobos are prefered (as they can restore all MP, and allow you bypass monster you don't feel like fighting).
Amongst Summons, Ifrit and Shiva are this, despite being more or less useless against enemies that are effected by their element.
As of late, Flans. Especially the XIII-2's incarnation of miniflans.