Even lampshaded in a flashback where Dio, when asked what he thinks is the weakest ability he's seen, states that in the world of Stands there are no objectively strong or weak ones.
For example, one of the most epic battles in the series is a rock-paper-scissors match between two people whose Stands don't lend themselves to fighting, the two of whom can't read each others' moves, and our hero, Rohan only wins because he cheats.
This isn't just a factor in martial-arts series. It can show up in Mons series like Yu-Gi-Oh!, Pokémon, Beyblade, and Bistro Chef (better known as Fighting Foodons in the States), and Cooking Duel series like Yakitate!! Japan. Essentially, anything that has individual competition, typically in a Tournament Arc, will tend to end up with this kind of arms race.
Displaying formidable multitasking skills, the eponymous Magical Girl Lyrical Nanoha is revealed in the series' ancillary manga to be able to train in the magical equivalent of VR constantly while leading her normal civilian life, apparently to explain her jump from neophyte magical girl to hardened battlemage in the short interval between the show's first two seasons. After the second season, she trains so hard she becomes known as "Ace of Aces" and is promoted to Combat Drill Instructor in her early teens. In the third season, another character trains herself nearly to the point of exhaustion in an effort to prove herself (but doesn't quite succeed).
It didn't work because the show decided to point out that mindlessly trying to improve yourself with brute force tactics doesn't work, it just breaks you down. Nanoha had a Heroic RROD in during the second Time Skip, specifically because of all the training she did.
And when Teana actually started listening in class, she got a lot better herself.
The StrikerS manga devotes two chapters to explicitly deconstruct this trope: the younger cast try to figure out the "strongest fighter" in their organization, but eventually learn An Aesop that no matter how strong your opponent is, he always has a weak spot which you can exploit to defeat him. The winner is thus determined not by a single variable but by a combination of quick thinking, experience, magical power, and sheer luck.
A decidedly more magical example, Negi trains constantly for fear of harm coming to his students. He's particularly fond of using Evangeline's time-warping villa to add extra hours (or rather days) to his day to train more.
Although the battles are of actual skills (usually) in MÄR, the good guys are often sent to train themselves in this manner for the next set of battles whenever the story allows, often using a time-warp ARM in the process.
In Love Hina, Motoko is stunned when Seta defeats her in martial arts, and becomes fixated on challenging him.
The titular character in History's Strongest Disciple Kenichi already goes through a daily regimen of Training from Hell to build up his stamina and improve his skills, but the few times he actually loses to an opponent, such as Odin and Tirawat Koukin, he forces himself to train even harder for the inevitable rematch.
The very reason Kyo of Fruits Basket goes into the mountains to train for six months is to one day beat his cousin Yuki in a fight (the manga reveals this is a lie covering up an unrelated Heroic BSOD, though). Every time Yuki wipes the floor with him, he only vows to become stronger... over... and over... again.
The 5 protagonist of Mobile Fighter G Gundam spend a good period of the middle of the series training in the Guyana Highlands in order to overcome individual failures (For Domon, it's his unfocused rage actually helping the Devil Gundam; for the other four, it's DG Cell possession making them question their own motives).
Mugen from Samurai Champloo always trains to the point of absurdity when there is an opponent stronger or more skilled than he is.
Swan is a shoujo version of this trope, but with ballet as the plot point.
Grappler Baki. The greatest styles are "Total fighting" (Baki) and "Every martial art you damn please" (Yujiro).
Benny of Black Lagoon quotes this almost ad verbatim during the Greenback Jane arc.
This phrase is actually said in the Marvel comic Immortal Iron Fist. The protagonist would necessarily have to have the strongest kung-fu if he could kill a dragon barehanded, so it wasn't empty boasting. Eventually he does get his ass handed to him by a guy who's even better.
Unsurprisingly, since it is a kung fu movie and owes a certain amount to anime tropes, Kung Fu Panda has this as the basic set-up for the fight with the Big Bad: that in order to defeat Tai Lung, the Dragon Warrior must attain the Dragon Scroll, for only with its cosmic powers can he hope to be strong enough to prevail. To hammer this point home (as if it needed it), after learning that his former student has paralyzed the Furious Five, Shifu declares that "Tai Lung has gotten stronger." Cue the delivery of the scroll to the understandably freaked Po.
Parodied in American Pie during an overblown speech by Kevin.
Revenge of the Sith: Obi-Wan says he is unwilling to kill Anakin, but Yoda sends him to fight him anyway, because "strong enough to face this Lord Sidious, you are not." Yoda finds he isn't strong enough either.
The basic premise of the Stephen Chow flick Kung Fu Hustle where everyone's kung fu is stronger than someone else.
In Man of Steel, Zod boasts during his fight with Superman that he has trained his whole life to be the perfect warrior. What he says has a fair degree of logic, though. Would you expect a person with little to no combat experience to stand a chance against someone who was specially trained to be a warrior?
In the Animorphs book The Ellimist Chronicles, this sort of arms race between Ellimist and Crayak ends up destroying a handful of galaxies and ripping a hole in spacetime that makes them omnipotent, universe-controlling gods.
Averted up the wazoo in The California Voodoo Game, where hypercompetent Bishop's martial arts techniques vastly out-class the limited training of Alex Griffin, yet Griffin still kicks Bishop's ass because he's so pissed off that he doesn't care if his ribs or fingers get broken: he just keeps smashing his opponent into the walls, denying Bishop the elbow room required for his dojo-ballet moves.
And their source material, the Super Sentai series, in a lot of cases, but especially in Juken Sentai Gekiranger, where the entire point of the series was two rival schools of fighting squaring off against each other.
In the finale of Dexter - season 3, there is a delicious example of this, in a scene with the season's "Big Bad" and Dexter. Remember Doakes's comment about why would a nerd like Dexter need to learn advanced ju-jitsu? Well it's so that when other killers tie up Dexter and are armed with a pretty menacing blade he can overpower them and snap their neck like a twig...with one good hand as he had to break his other one to get free from those pesky ropes that were holding him(Dex) down. Very strong kung fu indeed.
The X-Files: In "The Unusual Suspects" (origin story for The Lone Gunmen), Frohike is stumped by a mystery Byers and Modeski brought to him, and they end up on Langly's doorstep. Langly forces Frohike to quote this trope before allowing them in to use his equipment.
It's an actual character flaw, explicitly called this, in Deadlands. It's intended for Enlightened, but could thematically be used to create any character who takes great pride in their key fighting skill and is frequently fighting others to prove their superiority.
Most incarnations of the Street Fighter series portray Ryu as a loner who does nothing but train in his spare time. He is motivated to do this because Akuma killed his master (who was also Akuma's brother). Akuma himself has sacrificed his whole life to the way of the Satsui no Hadou ("Murderous Intent"), striving forever to become a stronger fighter, but also looking for someone who can defeat and kill him. He once defeats Ryu and then spares his life because he knows Ryu has the potential to become the greatest warrior on Earth (and the potential to fall into Satsui no Hadou as well).
The rift that developed between Aoi Matsubara and Sakashita-senpai in To Heart was the result of the latter's unacceptance of "Extreme Fighting" as a martial art.
This rift disappeared when Aoi won the Extreme Fighting Championship in To Heart: Remember My Memories when Aoi defeated her senior Ayaka Kurusugawa who had defeated Sakashita earlier on in the tourney with one roundhouse kick to the head, a la Chuck Norris.