(WARNING: This article contains spoilers for Game of Thrones Season 6 and all of the books.)
Game of Thrones is one of the richest TV shows of all time in terms of backstory. The book series that the show is based on, A Song of Ice and Fire, has thousands of years of history packed into them. However, when you adapt such a long story into a television series, you have to cut some parts out. There are many characters that play huge parts in the books that aren't even mentioned in the TV series. Let's take look at some of them.
(Note: This list will not include character's whose plot lines have been cut to make the show shorter, like Lady Stoneheart.)
3. Elia Martell
Even though we haven't seen this character on the show, you've heard her name many times. Elia is the sister of Oberyn Martell a.k.a. the Red Viper a.k.a. dude who got his head crushed.
Elia was also the wife of Rhaegar Targaryen, Daenerys's brother and Jon Snow's father (according to the show). Elia is the reason for Oberyn's hatred of the Lannisters, as Tywin ordered The Mountain to kill Elia and her children during Robert's Rebellion. This makes her the reason why The Viper volunteers to be Tyrion's champion in his trial by combat. That then leads to Oberyn's daughters and wife rebelling against his brother and taking Dorne.
The reason we probably won't see Elia in the show is because she's been dead for years and we would only be able to see her in a flashback or one of Bran visions. It's likely that the producers probably want to save as much money as possible, and since Elia is not quite as important as Rhaegar or Lyanna, she'll get cut first.
2. Ser Arys Oakheart
Arys is a more unknown character to people who watch the show, even though he has technically appeared. Apparently, he's often appeared in the background of the show, but has a more prominent role in the books.
In A Feast For Crows Arys appears in a few chapters. He is a kKngsguard tasked with protecting Myrcella Baratheon when she is in Dorne. In the books he is also Arianne Martell's lover. Arianne is the princess of Dorne and niece to Oberyn. Together, they hatch a plan to put Myrcella on the Iron Throne after Joffrey's death. However, after an attempt to get Myrcella away from Arianne's father — who doesn't want to start a war with the throne— Arys is killed.
The reason we probably won't see in Arys as a major character is because he's already appeared in the background multiple times and even though the Sand Snakes storyline is in progress on the show, Arys and Arianne are not featured.
1. Victarion Greyjoy
In the books, Victarion is the uncle of Theon and Yara (Asha) and brother of Euron, the current ruler of the Iron Islands. While we have seen Theon's other uncles (Euron and Aeron), we haven't yet seen Victarion.
According to the novels, Victarion attempts to become king of the Iron Island but loses the Kingsmoot to Euron. Victarion is then ordered by Euron to go to Mereen and fetch Daenerys so that Euron can marry her. Euron and Victarion have a complicated relationship and due to past events, Victarion instead goes to Meeren to marry Dany himself.
The reason we probably won't see Victarion is because the storyline he's featured in has already started without him. I could have just as easily picked his brother, Aeron, for this spot, but he has already appeared in the show a few times as a priest of the Drowned God.
BONUS: Aegon Targaryen
This is kind of a cheat pick because Aegon is most definitely dead on the show and even in the books there is doubt about whether the person called Aegon actually is whom he claims to be.
Aegon is the son of Rhaegar and Elia as well as Dany's nephew. In the books, Varys and his friend Illyrio claim to have saved Aegon in secret and raised him to be the perfect ruler. Aegon is going to take Westeros with a company of Sellswords by the name of The Golden Company. There are theories that Aegon is actually a pretender and not a real Targaryen. (There's a great video on this by Alt-Shift-X on YouTube.)
The reason Aegon probably won't appear is because it would be too late to introduce such a big twist in the main conflict. For the moment, it seems like everything is about Daenerys, Jon and Cersei. To introduce Aegon would muddy the water and confuse some viewers.