What has Selmy ever done to suggest he’s not every bit the badass he’s supposed to be? Has he ever been beaten? He’s over 60 by the time the books start and still nobody has been able to deal with him that I can recall.
I’d argue Littlefinger, because as we know from the Art of War
“For to win one hundred victories in one hundred battles is not the acme of skill. To subdue the enemy without fighting is the acme of skill.”
I vote for Chela, she has claimed 46 ears, after all. Or, in just one more year, Arya.
ETA: If we allow GO’s style of fighting, then I predict Tyrion will out-think Little finger in the end.
Drogo is definitely up there. The Dothraki only cut their hair when they lose in battle, and Drogo’s hair was past his waist, having never been cut. And his life was basically constant battle. That’s results.
Ned held his own against Jamie Lannister until he was attacked from behind by one of Jamie’s men. He may have won the fight without the outside interference.
Jamie dropped at least ten men before being captured and when we saw him fight after a year of being prisoner Brienne marveled at how incredibly fast and good he still was. And his hands were chained together.
I believe Brienne even thinks something to the effect of bloody hell, he would destroy me in a fair fight (at the same time as Jaime is thinking bloody hell, the wench is stronger than me). And Brienne is really good and really strong.
Barristan “The Bold” Selmy ranks above everyone else by virtue of the fact that he had great deeds before the story begins, like some others often mentioned as the best like Arthur Dayne - the Sword in the Morning – but Barristan is still alive and winning. His fighting was described by Jaime Lannister as watching “a painter who only used red.” He boasts that he could kill five armored Kingsguard as easily as “carving a cake” and everyone in the throne room believes him. When he was even formidable with a stick while disguised as Arstan.
Jaime had some great feats in his youth that gets him on the list. Ditto Drogo. Bronn, I think is the dark horse candidate that ranks well above what most people think of him. Bronn has no compunction about fair play, so he punches well above his weight class. In a fair fight, I don’t think he could take Jaime in his prime - but then it would never be fair. As much as I don’t care for him Daario is similarly gifted with murder.
I think people here are underrating The Mountain. In a Westerosi-style battle of armor and clanging swords he was rightfully regarded as invincible. Technically, he still won - or at least fought to a draw - in his final fight as he crushed Oberyn.
My cheeky answer is: “the best fighter in all of ASOIAF is Balerion.” 
I’d think Daemon Blackfyre would have to be up there, but I thought I read somewhere that GRRM said that Arthur Dayne with Dawn was the best one-on-one fighter in the world of ASOIAF.
Found the link: Just Dayne with Dawn vs. Selmy
To me, that speaks a lot of the sword “Dawn”. We keep hearing about it again and again in the series. I think it will become very important as the story plays out.
It is made from a falling star. A comet heralds the return of magic to this world…perhaps a connection there.
"Arthur: And now it begins.
Eddard: No. Now it ends."
“All knights must bleed, Jaime. Blood is the seal of our devotion.”
The “Battle for the Dawn” occurred when the Others last emerged.
Sword of the Morning - this isn’t a far stretch to connect as Lightbringer…is it?
Dawn brings light…
Sorry for the hijack!
Some modifiers to the zeitgeist rankings as they stand in the thread…(in no particular order)…
Syrio: He moves down three places. He is amazing but was only shown fighting rank-and-file guards, which are barely better than Power Ranger ninja extras. Simply put on some Yakkity Sax and he could defeat vanilla guards indefinitely.
Mountain: Up two places. An unstoppable force of nature, only an unfair trial by combat out of his element could bring him down.
Bronn: Up 1 place. He can take down any of the primary knights by exploiting their traditional training predictability. Maybe Robert Baratheon or The Mountain could take him down just because they combine power with unconventional fighting.
Brienne: Down 1 place. Formidable, but relies too heavily on her legs to reach a 15-round decision. She could easily be Mike Tyson-ed by higher-tier knights.
Drogo: Down 3 places. Dothraki acquire a lot of scars fighting among themselves with their silly curved swords and can horde like nobody’s business. But Drogo against even a mid-level armored knight would quickly turn into a pathetic game of “stop hitting yourself, stop hitting yourself”. Lemon cakes have more calories than horse hearts.
Final results: Tie for the top spot between Robert Baratheon and Barristan Selmy.
Wasn’t Prince Rhaegar Targaryen supposed to be quite the warrior… right up until Robert Baratheon smashed him with the warhammer in a long hard fight at the Ruby Ford on the Trident?
Surely being barely beaten by the best (or second-best) should get him a spot in the rankings?
I strongly disagree on the Mountain. He wasn’t “out of his element” - he had a choice of weaponry, and armor, as did his opponent. He is too stupid and predictable to be a deadly foe outside the chaos of full-scale battle.
He’s predictable, but I think he’s still pretty deadly, even 1 on 1.
I think that you have to take any reference to the Golden Boy with a healthy pinch of salt, filtered as it is through the perceptions of his adoring posthumous public. Barristan seems to think he was “able,” which is cool, but when Danaerys calls him a warrior without equal, or something like that, Barristan kind of coughs and says hey, there are lots of warriors out there, now.
Exactly what I was going to say. Selmy certainly didn’t think that Rhaegar was all that great as a fighter, and a battle he lost and winning the Tourney at Harrenhall are the only specific incidents I can remember.
I’d probably put him in the Jaime Lannister pile, solidly behind Selmy and Dayne.
I think maybe even a tier lower than that. The Cleganes and Jaime Lannister and Robert Baratheon fought for fun; he just felt like he had a duty to learn it.
Ha… I was about to compare him to Jon Snow and then I realized what I was doing.
Oberyn fought him like Syrio or Bronn would have - which is shown to be pretty rare in Westeros. Jaime at his peak would have been lucky to have even odds on The Mountain.
Forgive me if I misunderstand…but I don’t think Oberyn’s tactic would have been as effective versus any of the other knights out there. Heck, Bron could probably take Oberyn. But it doesn’t matter what style - the OP asks who is the best fighter in ASOIAF and I don’t think the Mountain really makes it. One on one he gets beaten. In a joust, or in a trial by combat. He is a giant of a man, true, but he wields that power so uselessly. He is a pawn in Westeros, and a pawn in a conversation about best fighter in ASOIAF.
Otherwise, Oberyn should be a top contender because he beat the Mountain, and I just don’t think he is.
Also, where does Jon Snow fall in this list? Certainly NOT that high, I should think.
What about fancy-pants-Mance “the King Behind the Wall” Raider?
Jon Snow moves way up if Ghost is around.
Mance will skipping the competition to have a jam session with Rhaegar; neither of those guys place much value on fighting prowess.
Where will Arya rank on the deadliest list by the end of the series? Never going to be a straight up fighting threat, but you certainly don’t want to be on her list.