I probably won't watch it a 3rd time but the reasoning I had for watching it a second time was because I was very confused about certain aspects of it even after seeing all of the episodes over the course of a 10 day period or so.
There will be spoilers moving forward
I normally will watch a 1 hour show while I am in bed and this could have been part of the problem because things like House of the Dragon are not of the "casual watching" variety. You really need to pay attention to know all of what is actually going on.
After watching the entire show the first time I had the general gist of things but I was kind of in the dark about a lot of it including and especially how the ascension to the throne came into question to begin with and how it is that certain members of King's Landing were a lot more responsible than others as far as stealing the throne was concerned.
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Take the above picture's characters for instance: I was a little in the dark about Alicent's involvement in stealing the throne the first time through and this is because you REALLY need to be paying attention to the scene where she witnesses the King's dying breaths and if you do see that a 2nd time you will understand that I did and that is that it is actually quite conceivable that she actually did believe that it was Viserys' genuine wish to see Aeogon ascend to the throne. However, it becomes evident to many, including her at one point during the small council meeting that there were some nefarious intentions at play long before she even mentioned those last words and she herself begins to have some regrets about what is being put in motion. I would imagine that we have not seen the end of her involvement in this and in season 2 I fully expect that she will be actively working against her own family.
The involvement of the other person in the picture, Sir Criston, has been scored and angry at Rhaenyra ever since hooking up with her many years ago and she pushed away his advances for the two of them to be together. I actually kind of find his movements against her family to be a bit much because it seems to me like given her station that she simply could have had him removed quite some time ago. I feel as though they show's makers have done a very good job of transitioning him from hero to villain.
Then we have the direct descendants of Viserys and Alicent: A group that I didn't really pay as much attention to as I should have the first time through. I think of all of them that I enjoy, at least the acting, of Aegon the most. He was a relatively background character for 80% of season once and he gets thrust into the limelight right at the end he starts to shine as a major player in an instant.
Basically all we are really told about him up to that point is that he is a bit of a deviant and a drunk and that is really the only insight we get into his life until episodes 9 and 10. Then we are shown that he is a very reluctant participant in his assumption of the throne and this will likely come into play later as I presume that the Hand and his mother will be making all of his choices for him as he does his best to simply maintain appearances in public. I think we will end up seeing a real conflict between the 3 with it perhaps culminating in a battle of sorts between the Hand and the Queen.
Halaena has played almost no role in the series at all as she is just the crazy sister that doesn't actually seem "all there." There was a scene towards the end of series one where she is in a room doing some cross-stitching of a spider (she has a fascination with insects) and there are 2 very blonde kids on the floor in her quarters. While it isn't explicitly spoken of, there is some indication that these babes might actually be the children of her brother Aegon because well, who else's would they be? It's not like they ever even spoke of her being betrothed or had a love interest of any sort. This could end up being a non-issue or it could be a major plot point. We'll have to wait and see.
Lastly we have Aemond, the child that lost an eye in a knife fight after claiming a rather large dragon whose name eludes me at the moment, just like the names of almost all of the dragons in the series. I like the portrayal of this character a lot less and he seems to be quite the brat and I guess that might have been the intention. I mentioned that there would be spoilers at the start of this but I will say it again: THIS IS A HUGE SPOILER THAT I AM ABOUT TO SAY!
When Aemond ends up killing Rhaenyra 2nd son "Jace" at the very end of the last episode of the first season you need to pay attention and for me it involved even having the subtitles on because the entire sequence is very loud and the action happens quickly. In the end I think everyone can tell that Aemond didn't actually intend to kill Jace but he simply didn't have control of his dragon and neither did Jace, whose own dragon actually made first combat contact.
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Then we have Daemon, a character that I didn't much care for until his turn of events that quickly lead to him being married to Rhaenyra. Prior to that change of events he seemed to be the Tyrion Lannister of this show in that he didn't much care for the "proper way of things" and didn't really have much aspirations of power that were his own. He accepted his brother being on the throne but just disrespectfully didn't really want much to do with it.
When his true fate of being with Rhaenyra is exposed at their wedding, his character begins to truly shine. He obviously is going to be an absolutely massive part of the story moving forward and I was quite pleased to see him move from being the brother who wants nothing to do with the royals to a ruthless strategist who everyone, including his own wife, need be very afraid of.
I didn't recognize most of these things when I watched through the series the first time through. I think I missed a lot of the key plot points because of the massive amounts of difficult-to-remember names that were getting thrown at us constantly and maybe I wasn't paying as close of attention as I should have. One of the biggest things that I missed was that during the coronation ceremony of Aegon when a dragon ridden by Rhaenys bursts through the floor and disrupts the entire thing. The first time I saw this I thought it was kind of hokey and convenient that she was able to get a hold of her dragon, don her armor that just happened to be nearby as well, and detonates the floor to threaten Alicent, Aegon, and Otto, then fly away without killing anyone of consequence. Looking at it a second time I actually really admire this scene because it shows that she has a tremendous amount of honor. She could have killed everyone on the stage there but didn't, because she understands the order of things.
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I didn't much care for her character for most of season 1 because she just spent a lot of time whining about how the throne was meant to be hers, as true as that claim might be it seemed to be a side-issue at best. Now, I think that her character and how she decides to align herself and her family could end up being one of the most interesting aspects of the overall story moving forward. I think at this point she has abandoned all hope of ever claiming the throne and out of all the players involves, she might be the one who is most interested in what is just, rather than being at all selfish about her own claims.
I am disappointed to discover that we have to wait a full 2 years before the next season will arrive. I may end up watching all of them a 3rd of 4th time to scratch that itch in the meantime. I will say this though: I would much rather wait and have a quality product than allow the writers to pump out some garbage just so they can make a timeline.... which is what I feel they did with the last 2 seasons of Game of Thrones.