Every good book series needs a villain right? Well, in the first few books it looks as if Ser Jaime Lannister is just that guy, that is until the end of a Storm of Swords and into A Feast for Crows. From there the reader's view of Jamie changes drastically. With that said its the Kingslayer himself, Jaime Lannister that makes my list as the #3 Greatest Warrior of the Westeros.
#3 -Ser Jaime Lannister, The Kingslayer
The Kingslayer is a tricky character. Like every character Martin writes there is more then meets the eye. Certainly at first glance we see a shallow, cruel man (throwing Bran out the window anyone?) But by book #3, A Storm of Swords we see a much more complex character.
Jaime is in every sense the golden boy, just look at him! He has everything and oh yeah he gets the hottest girl in the kingdom, too bad it's his sister Cersei. See that's where he goes really wrong in the character department, the things he does for love, namely pushing Bran out the window in Winterfell.
First off let start with his nickname: The Kingslayer. He earned this moniker when he slew the Mad King Aerys II during the sack of King's Landing when Aerys tried to destroy the city with the help of wildfire. For this act Jaime thought people would love him for saving the city, but they were in fact recoiled by him. He remarked about this fact saying that he wanted to be Ser Arthur Dayne, but became the Smiling Knight, referring to the previous generation's monster, i.e. the predecessor to Ser Gregor Clegane.
Despite this Jaime is brave, so brave that as a squire for Sumner Crakehall he saved his life when he stopped Big Belly ben, another member of the infamous Kingswood Brotherhood along with the Smiling Knight. It was after this that he was knighted by Ser Arthur Dayne, the Sword of the Morning himself.
Further intrigues involving his sister led him to become a member of the Kingsguard as a means to be able to stay close to his sister as well as void marring Lysa Tully. Knighted by the Lord Commander, Gerold Hightower, Jaime took the white, much to the displeasure of his father Lord Tywin. It's also important to note that at 17, he became the youngest member ever to serve in the Kingsguard.
In combat Jaime is a master with the blade; he can make his sword do whatever he wants it to. His sword abilities make him one of the deadliest warriors in the Seven Kingdoms. He is one of the younger characters to make the list and represents the generation of Ser Loras Tyrell, and his brother Garlan. In the lance he is quite skilled as well having won tourneys that he reflects upon when he becomes Lord Commander.
In battle he is fierce as befits the words of House Lannister: "Hear me roar!" He defeats Ser Edmure Tully at the Goodtooth and then leads his host onwards towards the Riverlands. He is only bested by Robb Stark at the Battle of the Camps through superior numbers in which he is captured. In the process of trying to get to the Young Wolf he slays several men including the Karstarks. from there he is taken prisoner and finally only released by Catelyn Stark. So for that he gets bonus points for not only being a great warrior, but a leader of men as well.
Jaime's only real blindside is his devotion to his sister. Their relationship is so lopsided that he is at a handicap, until their falling out in A Feast for Crows that is. It is also in a Feast for Crows and after his sword hand is cut off that Jaime becomes a different man. Vargo Hoat does this in order to drive a wedge between the Lannisters and Roose Bolton, but this ploy fails. His hand being severed is nothing more then a gambit in the Game of Thrones and achived none of the aims it was intended to convey. Jaime's whole life is wrapped up in how good he is at arms. It's when he can't swing a sword he drops into a depression and only emerges after he saves the life of Brienne of Tarth.
Jaime's reputation is as such that he is able to get Ser Boros Blount (when Jaime can hardly swing a sword on his return to King's Landing) to acquiesce to his commands. Such is his reputation that he is able to humble him with a member of the Kingsguard with words alone, knowing full well that people know of the reputation of the Kingslayer not as he is now.
Still later in a A Feast for Crows, Jaime begins the oft-painful process of sparring with Ilyn Payne in order to learn to swing a sword with his left hand.
Prior to his sword hand being cut off the Kingslayer is one of the deadliest warriors and as such merits his inclusion on this list. It's well earned and only time will tell if Jaime is as good with his left hand as was with his right.