The existence of bending in Avatar: The Last Airbender and The Legend of Korra means that most serious fights in the world involve the manipulation of one of the four elements (or all four if the Avatar is involved.) The three nations, excluding the Air Nomads, employ their respective benders in most fields, such as the military, industries, architecture, and so on.
That being said, how would the best fighters in TLOK be ascertained when not considering bending powers in the equation? In other words, what happens when two characters are thrown into battle and made to fight each other like nonbenders. Can the strongest and most unique benders in the world earn a name for themselves with nothing more than unarmed combat at their disposal?
Tenzin
Tenzin is one of the strongest fighters in the world, showing himself to be as strong and as agile as many villains. In his battle with Amon's minions, he easily avoids their attacks (continuing to do so even after he is backed into a corner.)
Tenzin singlehandedly takes on Zaheer, Ming-hua, and Ghazan, and it's evident that he can take quite a significant amount of punishment from their assaults without slowing down.
Tonraq
Korra's father, Chief Tonraq of the Southern Water Tribe, is an inimitable Waterbender, but the fact remains that his military knowledge and incredible aptitude in terms of physical weapons make him a far more dangerous opponent than most benders.
This is shown when Tonraq takes on Zaheer in order to rescue his captured daughter, only failing because of his opponent's overwhelming talent.
Suyin Beifong
Suyin, Toph Beifong's younger daughter, is revealed to be the "leader" of Zaofu, a purely Metalbending autonomous city within the Earth Kingdom. Her introductory dance shows that she is amazingly nimble and deft with her movements, something she blends into her unarmed combat.
Despite being an accomplished Metalbender, Suyin depends on her dexterity to avoid bending attacks, like when she has to face P'Li, one of the most dangerous benders in the world.
Bolin & Mako
Brothers Mako and Bolin exhibit distinct fighting styles, but it can be safely assumed that they are around the same level when it comes to pure skill. Mako's speed and strength are admirable, explaining why he is a rising star in Republic City's police force.
On the other hand, Bolin's battle expertise is built around his Earthbending, which means he prefers to employ techniques like tackling if he needs to quickly neutralize an opponent. Also, the brothers are likely to agree that they are similarly talented.
Kuvira
Leadership and charisma aside, Kuvira is an exceptionally gifted fighter even when she's not making use of a single ounce of her prodigious Earthbending genius. She has regularly proven herself to be a capable combatant, especially against the Avatar, although it must be admitted that Korra is held back by her own overwhelming internal demons at the time.
One of the more impressive facts about Kuvira, however, is that she emits an aura so intimidating that most of her opponents back away from any hint of potential battle.
Amon
Amon pretends to be a nonbender fighting for the rights of his people, whom he rightly claims are ignored and downtrodden by the bending elite of Republic City. Of course, his facade is vaporized when Korra unmasks him as Noatak, the son of the criminal Bloodbender, Yakone.
In any case, Amon doesn't use his bending in battle, choosing to rely on his spectacular martial abilities (not to mention the art of Chi Blocking as invented by Ty Lee in the prequel series.) Shocking as it sounds, he can effortlessly recover from Lightningbending attacks in the span of seconds.
Lin Beifong
The older daughter of Toph, Lin Beifong is the Chief of Republic City's police, a career choice that requires her to sharpen a lot more than just her Metalbending.
She is a capable fighter, as seen in her battle with her baby sister, regardless of how tired she is from her recent acupuncture experience. Lin can not only endure attacks from chi blockers but also manage to defeat a horde of them simultaneously.
Zaheer
Zaheer may have grown into a masterful Airbender and learned how to fly, a technique that only two humans in history have been capable of (neither of them being the Avatar.) But his role as the leader of the Red Lotus begins when he's a nonbender, proving that he could take down those technically more powerful than him.
As such, Zaheer is considered to be such a high-risk prisoner that he is kept trapped on a mountain top for over a decade to prevent him from escaping. After developing Airbending, it takes an entire platoon of Airbenders to bring the villain Zaheer down.
Asami Sato
Korra often states that Asami never needs to actually enter into battle because the Avatar will always be there to protect her girlfriend. Romantic overtures notwithstanding, Asami is not one who depends on anyone else for support, nor does she need to.
She fights on par with opponents as dangerous as Amon's Lieutenant and his electric kali sticks, and her childhood training comes to the forefront in nearly all of her battles. Asami armed with an electric glove is a nightmare for her enemies.
Avatar Korra
There is absolutely no doubt that Korra is far superior at nonbending combat than everyone else on the series (or, to be honest, the entire franchise.) She is supremely strong, being able to carry objects far too heavy for most grown men to even move an inch, let alone send flying.
Korra tends to use her legs more than her arms, efficiently defeating her opponents with borderline impossible kicks and punts. In fact, it can be argued that the Avatar would be able to dominate the strongest benders without using any elemental powers of her own.