
It's hard to tell which of my Game of Thrones opinions are the most unpopular, but I'm guessing one of them is that I don't like Jaime Lannister and I never have. There are a few reasons why I hate him:
1) He tried to murder a child: During the first episode of the show there's a kid named Bran Stark who accidentally sees Jaime and Cersei Lannister having sex. Cersei Lannister is married to the King, who is not Jaime. Jaime throws Bran off a tower. He's essentially trying murder Bran in order to keep him from telling anyone about their affair. Let me repeat: Jaime Lannister tried to murder a child in order to cover up his affair. That's monstrous behavior no matter how you look at it. Jaime never expresses remorse for this in either the show or the books.
2) He's having sex with his sister. And it indirectly kick started a massive war: First of all, incest is absolutely disgusting. Second, his affair with his sister indirectly led to the war that happened during the next two books / seasons. What kick-starts the war is that Ned Stark finds out that Joffrey and his siblings aren't children of the king. Ned eventually learns about Jaime and Cersei's affair. Ned told Cersei that he knew about her affair. Cersei refused to back down and leave with her children, the Lannisters seized control and Joffrey had Ned Stark executed for treason.

Eventually Stannis and Renly Baratheon, the younger siblings of King Robert, find out about the incest affair. Stannis proclaims that he is the rightful king (to be fair, he is since he's Robert's oldest brother so naturally he would be the rightful king) and refuses to bow to Joffrey. So now we've got claimants competing for the Iron Throne. Plus the truth behind Joffrey's existence led to Ned's death which is what caused Robb and the rest of the Starks to declare war on the Lannisters.
If Joffrey didn't exist, then Ned would never have found out and Stannis would have probably gotten the Iron Throne instead of fighting for it in a war. Considering all the major consequences from Jaime's affair I'd say it's good enough to consider his irresponsible affair a major reason why the war even happened. Obviously I feel the same way about Cersei too. It would have cost him nothing to not have sex with his sister, but he did anyways and now there's a war for succession.
3) Jaime Lannister is a Rapist: Here comes the most unpopular opinion regarding Jaime. In Season 4 he rapes Cersei shortly after Joffrey dies. He rapes his sister in front of their dead son. That's all kinds of screwed up.
Now this happened in Season 4, back when the show was still adapting the books. So the scene I'm talking about was the show adapting a scene that Martin wrote. The popular consensus among the fanbase is that the book version of that scene was consensual. I disagree with this.
My unpopular opinion here is that even the scene written in the book is actually rape based on how it's written. Jaime tries to have sex with Cersei, she says "Not Here", but he still has sex with her anyway. Afterwards, she says "that was folly". If a woman says "Not Here" I would take that to mean that she hasn't given consent. She later says "that was folly" and the book brings up other instances during that scene where she's obviously protesting and shows signs that she doesn't want to have sex with him. The only reason why she didn't get traumatized is because Martin didn't consider that a rape scene so he didn't write her getting traumatized from it.
Let's put a pin on this for now, but I plan on coming back to this point when I criticize the books. One criticism that I have about the way the books are written has to do with the way Martin writes rape and other sex scenes. This is a prime example right here and the fact that there's plausible deniability doesn't help his case either.
If this wasn't bad enough, there's still one more incredibly screwed up element to all of this. The show-runners didn't depict that sex scene as rape on purpose. Those idiots actually thought they were portraying consensual sex. Holy crap! Where do I even begin with that? Everyone else who saw that scene and their mom knew that was rape! How in the flying heck did the show-runners seriously think that was consensual? That is super messed up!
Other Popular Game of Thrones Characters I Don't Like
I'm going to do a rapid fire session because the sooner my anti Game of Thrones posts finish, the sooner we can all start gushing Avatar: The Last Airbender.
Theon Greyjoy
For the sake of time I'm going to copy / paste a bit about Theon from a Reddit post that explains him quite well:
"Theon essentially spent his entire life as a hostage and torn between two lives, two identities, two families, and two cultures with the ever present threat that he may be killed if his father decides to rebel again.
His blood family scorned and mocked him for his lack of Ironborn upbringing. He was forced to choose between one family he could never be a part of and the another family who mocked him. He chose wrong. He picked blood over the Starks and did whatever he could to sever any links he had to the Starks.

He wanted to prove that he was Ironborn and a Greyjoy of Pyke in the most extreme way possible. He did horrible things out of a deep and desperate desire to belong and to prove himself to his family...
The tragedy of Theon Greyjoy is that he had the family he always wanted but didn't recognize in time and no one ever told him. He was a Stark but not by blood or by name, he was brother to Robb, and a son of Winterfell. He picked the wrong side, the wrong culture, and the wrong family"
(Source: https://www.reddit.com/r/asoiaf/comments/2ex2gj/spoilers_all_what_is_your_opinion_on_theon_greyjoy/)

One of the "horrible things" Theon did back when he betrayed the Starks was murder two innocent children so that he could pretend that he held Bran and Rickon Stark hostage. That's when I lost all my sympathy for him. People felt sorry for him because he eventually got tortured by Ramsay Bolton. By "tortured" I'm including the fact that he got castrated and eventually Ramsay broke him. The thing here is that if Ramsay Bolton had simply beheaded Theon instead of torture him I would have been okay with that. Theon murdered children. I'm not forgiving him for that.
I also don't sympathize with him for wanting to be part of the Greyjoy family. He should have figured that they hated him and that he would have been better off with the Starks than the Greyjoys.
The Hound

Why I Don't Like Him: In the books he comes really close to raping Sansa Stark. By "really close" I mean that he holds a knife to her throat. He also murders a child and then makes a joke about it. In the show a lot of his dialogue can be boiled down to "F*** this. F*** that." What a swell guy.
Bronn

Why Do I Hate Him: In the books (and I think the show, but I'm not sure on that) he brings up the fact that he would murder a baby if he were paid enough to do it. Thankfully he never does in either series but the fact that he would openly admit to doing such is enough to make me despise him. Remember how I brought up when Book!Tyrion had a singer boiled in stew? Guess who did Book!Tyrion's dirty work? Bronn did.
Bronn is one of the few characters who survives all the way through the show and there really isn't a good reason why. He doesn't actually do much from Seasons 6 - 8. Somehow this guy winds up with a position of leadership at the end of the show. Bronn ends up with the title "Master of Coin" which means that he's in charge of the kingdom's money. The problem here is that Bronn has done nothing to earn this position. It feels like the show kept him alive as an act of fanservice.
Tywin Lannister

For those of you who haven't seen the show, Tywin is Tyrion, Cersei and Jaime's dad.
Why I Hate Him: Here's a list of stuff this monster did:
1) Ordered his soldiers to gang rape his son's wife: The only reason why he did this was because he was pissed off that his son Tyrion was marrying a commoner.
2) He lied to Tyrion, telling him that his wife was a "whore" who was paid to have sex with him: Tyrion eventually learns the truth that his wife actually did love him and that she wasn't a prostitute. In the books, this revelation is what causes Tyrion to kill his dad.
3) He openly admitted to Tyrion that he thought about killing him when he was a baby: Tyrion's mom died in childbirth and Tywin hated Tyrion for that. Also, Tyrion's a midget and Tywin hates him for that too.
4) Tywin knows that Tyrion didn't kill Joffrey but puts him on trial for it anyway and plans to screw him over: Tyrion wouldn't have faced the death penalty. He probably would have been sent to the Night's Watch before he decided to ask for a trial by combat. Still, a father putting a son who he knows is innocent on trial is a monstrous behavior.
5) He helped orchestrated the Red Wedding: One of the major reasons why Walder Frey even carried out the Red Wedding to begin with is because Tywin Lannister offered to protect him if he did. Tywin is considered to be one of the main orchestrators of that slaughter. Tyrion calls him out on this and Tywin actually tries to justify the slaughter.
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Final Words: Like I said before, there's a lot of awful, monstrous characters in A Song of Ice and Fire / Game of Thrones. John Barkmeyer hit the nail on the head: These characters aren't shades of gray. They're shades of black.