If you don't like fairytales or the fluffy sillyness of some fantasy, and you like drama like Breaking Bad, probably you'll really enjoy Game of Thrones.
It's all about a gritty, well-developed world close to the reality of what the middle ages were like, without clear lines between good or evil, with complex factions made up of many well-realised believable characters, with some fantasy elements added sparingly to increase the unpredictability and mystery. Here's a good quote from book author and (in some seasons) TV show exec producer George RR Martin:
a lot of the fantasy of Tolkien imitators has a quasi-medieval setting, but it’s like the Disneyland Middle Ages ... what I want to do is combine some of the realism of historical fiction with some of the appeal of fantasy, the magic and the wonder that the best fantasy has.
As much as I love historical fiction, my problem with historical fiction is that you always know what’s going to happen... Fantasy, of course, doesn’t have that constraint... what’s gonna happen next? I love this character, but god, is he gonna live, is he gonna die? I wanted that kind of suspense.
I'm not a fantasy fan, and personally find even popular fantasy like Lord of the Rings too "fantasy" for me, but I rate GoT as one of my favourite shows. When it does include fantasy elements, it's famous for subverting them. For example:
It doesn't lean on one-dimensional fantasy races of human-like being like orcs, elves, etc, who are all basically humans but with one motivation and one personality type. For example:
- One character is a dwarf, but instead of being from a "magical race of dwarves", he's simply a normal guy with dwarfism - a very well realised character dealing with all the prejudices and challenges that a well-born man with dwarfism would face in the Middle Ages, alongside his other agendas.
- There are giants, but these have a small role mostly as part of an interesting study in how skeptical, realistic people in the Middle Ages would respond to finding out that legends about giants actually had some truth in them.
- Dragons are presented as hard-to-train unpredictable animals, and no-one entirely understands them - again it's quite a believable portrayal of how people would react to a species they're heard legends about. Human-dragon interaction is closer to Jurassic Park than it is to, say, Martin Freeman chatting with Benedict Cumberbatch across a sea of CGI in the Hobbit (and is much less silly than Jurassic World).
- There are undead (or at least, something resembling undead), but their motivations (if they indeed even have motivations) are a slowly unfolding mystery. Not wanting to give too much away, I see this aspect of the show as a subtle but rather brilliant study in how incompetently and selfishly the majority of people and, particularly, politicians deal with a creeping threat that should, if there was any rationality in the world, unite them to put aside their petty power games and join a common cause. More XCOM than Independence Day.
- There is one faction (Thenns) who appear late in the show, who annoyed me as a non-fantasy fan a little. Strictly speaking they're more of a tribe than a race, but they edge a little bit close to being one of those credibility-stretching fantasy races where they all look the same, act the same and their motivation is "they're just like this". But this isn't nearly enough to spoil the show - they're just one small-ish faction out of something like 30 - and it's the only example of this I can think of.
- Magic is virtually never used as a get-out-of-jail-free / Deus Ex Machina plot device. It exists, but as a mysterious force no-one quite understands (I've not seen more than a few snippets of Lost, but I get the impression it has similar "is this supernatural or isn't it?" mysteries). A small number of characters periodically try to use it: without giving anything away, it often has unexpected consquences, it often doesn't pan out as planned, and some characters turn out to have been borderline-charlatans winging it, not really understanding what they were doing. No-one gets out of a pickle by waving a magic wand. There's at least one magic-assisted killing, but it's more of a plot twist which creates many loose ends, rather than a plot device to escape a predicament. I might even go as far as to argue that the way Game of Thrones uses magic is further from how magic is used in most conventional fantasy than plot devices used in many non-fantasy shows, like Sherlock's "mind palace", or Breaking Bad's Walt saving the day with a chemistry trick. If you can deal with those, you can deal with this.
- Characters are realistically portrayed. There are knights, and their armour is often shiny, but they're more like real humans given power, authority and status than "knights in shining armour". Some act like spoiled celebrities, some are cynical and disillusioned, some are stern professionals, some are schemers - all have convincingly real personalities and back stories that make sense. There's a fierce warrior woman introduced in a later season - but instead of dancing around making wisecracks in impractically skimpy armour, she's a realistic, serious-minded professional who turned to this career because her unusually butch physique caused her to be rejected from conventional high society. There are prophecies and prophetic dreams - but they're confusing and are misinterpreted by the troubled characters who have them as often as not, creating more problems than they solve. And so on - most fantasy tropes have at least one character who artfully subverts them into something believable and credible.
- There's lots of underhand politics and dialogue-driven drama. I'd say the ratio of fantasy action to family drama and politics is similar to the ratio of gangster action to family drama and problem-solving in Breaking Bad. It's not uncommon to go episodes with almost none, but when it does happen, it's intense, gripping and unpredictable. A major theme of the show is a tangled web of factions and powerful families pursuing many political agendas at once, and the quality of the political intrigue is very high - it's a thousand times closer to, say, House of Cards or Borgen than the paper-thin "I seek an audience with the Elf King" stuff typical of many fantasy shows.
- It's unpredictable. Good guys don't get their way unless there's a good reason why they should get their way. It's famously brutal with its main characters and main storylines - main characters die with little warning, prominent storylines can suddenly get snuffed out by events and just end dramatically, mid season. More so than any other show I can think of, actions have lasting consequences, and if good people are in a situation where, in real life, bad things would happen to them, bad things will happen to them. In this respect it's the very opposite of fairy-tale.
Watch the first episode of season 1 - it has maybe slightly more fantasy elements in it than a typical episode (especially in the very first scene, which is quite untypical), but otherwise is fairly typical in terms of tone and the mix of dialogue, action and plot twists. If you have no problem with the splash of fantasy in the first episode, you're unlikely to have any problem further on.