They become super overpowered very quickly (Dany) and can bore or agitate the reader by doing so.
So how does one write a proper Khaleesi who is still relatable, like-able, and on (maybe not totally) equal playing field? Do you give his/her dragons limitations? Weaknesses? Do you give him/her limitations? Weaknesses?
EDIT: Gasp I’m being attacked for having an opinion! D: Please note I love Daenerys and she’s probably one of my favorite characters (more in the show than the books though) in ASOIAF. But I can see she has a very loud and large group of haters who have valid reasons as to why they don’t like her or think of her as flawed. Yes, I am aware that Dany cannot control her dragons (completely), and yes I am also aware that she faces dilemmas and conflicts. But honestly? Compared to the other characters, Dany gets many of the things she wants/needs VERY early on in the story. She’s also dealt a very good hand in life being a Targaryen and all, not to mention how many times she’s been so incredibly lucky, it seems too fantasy.
Her Dragons definitely make her over-powered, not just in the aspect that she can decimate her enemies very easily, but everything she’s acquired (Unsullied, Khalasar, Slavers Bay, Ships, etc) has been because she had dragons. So to say Dany is not super powerful because she faces moral and internal dilemmas and conflicts is not 100% truthful, as you can clearly see it’s very easy for her to handle said issues because, Dragons.
P.S. : I love Dany, as I said before, but she progressively got EXTREMELY arrogant and entitled. Almost as bad as Viserys. Not to mention in the books, she is almost as sadistic and cruel as Cersei or even Areys. She literally has a man and his daughter tortured, but makes the dad watch first. Or just how sickening her lines about smiling every time she watched an enemy burn truly are. I very much love show Dany, but book Dany is a freaking psychopath (that also has dragons).
In the show, while season sevens war plot was extremely dumb (Tyrion and Dany have never been THAT incompetent, not even in Slavers Bay), they did a good job at humbling her and making her an interesting character again. After loosing Viserion, and nearly losing Drogon and her life on the Gold Road, she’s vulnerable and her dragons are not the unstoppable novelty that they were. The other side now also has a dragon, which helps level the playing field. While the entire way they went about the war was retarded (as Dany should have really just sacked kings landing with everything like everybody but Tyrion told her to), they did knock her off her high horse and re-opened her for growth as a Character.