"There was a boy called Eustace Clarence Scrubb, and he almost deserved it."
Eustace is portrayed at first as arrogant, whiny, and self-centred. It can be gathered from Eustace's behaviour, and the tone that Lewis used in describing his family and school, that Lewis thought such behaviour silly and disliked it a great deal. In fact, at the beginning of The Voyage of the Dawn Treader, Lucy and Edmund dislike visiting him and his parents, though that has mostly to do with Eustace's arrogant and unfriendly attitude. However, in the later books, Eustace is shown as an altogether better person, becoming a hero along with Jill Pole. It is mentioned in the Silver Chair that Eustace is afraid of heights, causing him to overreact when Jill goes too close to the edge of a cliff. In trying to stop her he falls. In other respects Eustace displays great courage and a fair degree of discernment in facing the challenges that confront him in the Narnian world.
According to Lewis's Narnian timeline, Eustace was born in 1933 and is 10 years old when he appears in The Voyage of the Dawn Treader; and by The Last Battle he is 16 years old.
We meet Eustace at the beginning of The Voyage of the Dawn Treader with the memorable opening line, "There was a boy called Eustace Clarence Scrubb, and he almost deserved it."
He is the only child of what Lewis describes as "very up-to-date and advanced people," who send him to a progressive mixed school. Eustace calls his parents by their first names (Harold and Alberta); his parents are vegetarians, nonsmokers, teetotallers, pacifists, and wear a special kind of underclothes.
Much of the narrative of The Voyage of the Dawn Treader concerns the personal growth of Eustace, as he is drawn into Narnia and aboard the eponymous ship along with Lucy and Edmund, and into adventures that bring him to realize how self-centred his attitudes are. Part of the story is told with extracts from his diary, where we see how skewed his point of view is. He describes the ship sailing in perpetual storm (though the weather is fine), and portrays the others as foolishly denying the supposed rough seas and refusing to face the "truth" of the situation. He complains when Lucy is given Caspian's cabin, and comments to the crew that giving girls special treatment is actually "putting them down, and making them weaker". Moreover, he cannot accept that he is in the Narnian universe: he imagines that he can "lodge a disposition" (or "bring an action") at a British consulate or a British court; and he is beaten by Reepicheep for treating the mouse as one might a circus animal.
Eustace wanders off by himself when the ship puts ashore on an unexplored island. He falls asleep on a dragon's hoard and finds himself transformed into a dragon by "greedy, dragonish thoughts" in his heart (cf. Fafnir). Worse, he is now in constant agony from Lord Octesian's arm bracelet, which he put on as a boy but is too small for a dragon's leg. Upon return to the Dawn Treader, he is nearly attacked by the crew until Lucy asks if he is Eustace, to which he vigorously nods his head. Being a dragon changes Eustace; instead of behaving like his usual sulky self, he helps the travellers find food, shelter, and a tree to serve as a new mainmast. The problem comes when it is time to leave the island, as the ship cannot hold or maintain a dragon. Reepicheep displays sympathy to Eustace's plight despite the boy's prior cruelty to the mouse and they eventually become friends.
Eventually, Eustace meets Aslan, who returns him to human form by peeling off his dragon skin and sending him into a refreshing bath. Edmund shares with Eustace his own redemption story, observing that "you were only an ass, but I was a traitor." Eustace improves after this, though he still exhibits bad habits - "he began to be a different boy" is how the author puts it - and becomes a valuable member of the expedition. When the ship is in danger of being sunk by a giant sea-serpent, Eustace boldly if ineffectually attacks the monster, using only a sword. When Eustace returns home after his adventures, his mother thinks he has become tiresome and commonplace, blaming the change on the influence of "those Pevensie children"—though everyone else thinks he has become a much better person.