While Game of Thrones included some big Targaryen twists concerning the likes of Jon Snow and Daenerys Targaryen, it decided to cut the most surprising Targ-related storyline from George R.R. Martin's A Song of Ice and Fire books (so far): the reveal that Aegon Targaryen is still alive. The second child and only son of Rhaegar Targaryen and Elia Martell, it had long been assumed that Aegon had been brutally murdered by Gregor Clegane, aka The Mountain, during the Sack of King's Landing towards the end of Robert's Rebellion, alongside his sister, Rhaenys. A Dance with Dragons, however, revealed that wasn't the case, with Tyrion travelling with someone called Young Griff, later confirmed to be Aegon Targaryen (or at least, that's how he's presented to the reader), who had been saved from King's Landing, and raised to be the perfect ruler of Westeros, with his eyes firmly set on the Iron Throne.
It's a huge turn of events that sets up even bigger things to come, putting Aegon/Young Griff on a collision course with his aunt, Daenerys, given both are after the same thing. And yet despite that, it is completely absent from Game of Thrones. Tyrion's long journey is massively expedited, most of the characters from this storyline are absent, and there's not even an outside chance or rumor of Aegon's survival. The Targaryen drama is instead saved for Daenerys and Jon Snow, but why was the character who could become Aegon VI cut from the show?
The most obvious reasoning behind Game of Thrones cutting Aegon Targaryen is that it came too late in the day. If the show were to include him, then it would've had to introduce him during season 5, which marked the point where they started to veer away from the books in more pronounced ways than ever before. Showrunners David Benioff and D.B. Weiss had always envisaged the series as being around 70-hours and seven seasons, and that meant they decided to cut a swathe of the later characters to maintain focus on the established ones. Some of this streamlining was necessary, because anything approaching a completely faithful adaptation would've been overwhelming to manage given the sheer amount of characters involved, but losing major plot lines and characters such as Aegon, Arianne Martell, and in a different part of Westeros, Lady Stoneheart, was an unfortunate consequence. Aegon alone would've meant several more episodes at the very least, and that expands once you factor in the supporting players.
It's also quite clear to see which characters Game of Thrones decided to keep its focus on instead of Aegon. Cersei Lannister claimed the Iron Throne and, in Game of Thrones season 8, faced off with Daenerys, which resulted in the latter destroying King's Landing atop her final dragon, Drogon. This storyline is one that A Song of Ice and Fire is headed to as well, but in a different way: there it'll be a second Dance of the Dragons, as Dany faces off against Aegon (albeit under different circumstances, and likely at a different point). Since Cersei had already been built up as a key villain, and one of its most complex characters, then the showrunners decided to stick with her instead.
Similarly, Game of Thrones had another major Targaryen twist to build to: the reveal that Jon Snow was himself a Targaryen - the secret lovechild of Rhaegar and Lyanna Stark. While many had guessed this long before the moment it was finally announced in the show, thanks to the hugely popular R+L=J fan theory, it was still intended to be a real twist. One reason Game of Thrones cut Lady Stoneheart was because they had Jon's resurrection coming up, and didn't want too many characters coming back from the dead to lessen its impact. It's highly likely, then, that the same logic factored into cutting Aegon Targaryen, a decision that would've been made around the same time as Benioff and Weiss geared up for the endgame.
Alongside all of this, the omission of Aegon Targaryen from Game of Thrones can be taken as further evidence that he is a fake. A popular theory surrounding the character is that he's not really a Targaryen - with fans dubbing him "fAegon" - and is most likely a Blackfyre pretender. This can work in the books, where there's much more room to explore this, and it caters to a different audience. But no doubt the showrunners, who would've been told of Martin's plans for Young Griff, felt that introducing a character as a new Targaryen in season 5, only to reveal he wasn't crucial to the endgame - which, as expected, came down to Jon Snow and Daenerys - might've been overkill. It's a shame in some ways the story wasn't adapted, but it does mean it's yet another aspect where fans are unspoiled ahead of the final two books.