Kenzo Tenma, brain surgeon supreme, is one of these in Monster.
Lelouch and Nunnally from Code Geass, after the former falsely claims their deaths in the Britannian/Japanese war that resulted in the colonization of the latter, seeking refuge with the Ashfords to avoid being used once again as political pawns by the Britannian Empire. In R2, Nunnally is readopted into the family (as a pawn against Lelouch), and Lelouch is left alone.
Princess Arika and Princess Theodora of Mahou Sensei Negima! during their time with Ala Rubra, and thanks to later events and revelations revealed after the Ostia Tournament, we could add Arika's son, Negi here as well.
King Van of Fanelia from Vision of Escaflowne spends most of the series on the run from the Zaibach empire, who need him so they take control of the Escaflowne themselves. While the royalty of countries like Austuria are at least somewhat sympathetic to his plight, they're not entirely willing to risk their asses by truly granting him refuge.
Princess Leia in Star Wars spent years in hiding, running away from the Empire. This is a subverted trope in that she was already a member of the Rebellion before she went into hiding and going to them was logical after being captured at the start of A New Hope, possibly even if Alderaan had not gone kablooey. In fact, that may have been where she was headed even before Vader thought it'd be fun to sack the Tantive IV.
Her mother and adoptive father also fit this trope by the end of Revenge of the Sith. The Emperor was in control of the Senate, forcing them to go into hiding. Of course, Amidala tragically did not survive long.
Subverted with Maladict in the Terry Pratchett novel Monstrous Regiment. When Maladict joins the army, the recruiting officers are wary of what would make a gentleman sign up, and Maladict immediately assures them there isn't a price on his head (before they get round to asking).
Alan Breck Stewart in Kidnapped by Robert Louis Stevenson.
The Silmarillion is full of these. Beren is probably the cleanest-cut example; his father's kingdom gets overrun by Morgoth's army, and he lives off the land rather than leave it, first as part of a group of twelve warriors, then alone after the group gets betrayed. There's also Túrin, Elwing, Eärendil, and probably a bunch of others that I'm forgetting about at the moment.
In The Chessman of Mars, U-Thor, after he questioned his jeddak's injusitce, had to fight his way to freedom and then escape.
Saukendar (Shoka) in C.J. Cherryh'sThe Paladin, once teacher and adviser to the boy Emperor, who flees when his friends are framed for treason and lives in exile just beyond the Empire's borders.
The Scarlet Pimpernel: The de Tournays and the other noble émigrés rescued by the Pimpernel.
Paul Atreides and his mother in the original Dune seek refuge among the Fremen after the Harkonnens kill his father the Duke Leto. Whom he rallies into an unstoppable army and uses to claim the Golden Lion Throne.
In the prequel novels, the entire House Vernius goes renegade after their world of Ix gets invaded by the Tleilaxu with the secret support from Padishah-Emperor Elrood IX and, later, his son Shaddam IV. All but one end up dead before Rhombur Vernius re-takes the planet with the help of the Atreides and receives his father's title of Earl.
In Sergey Lukyanenko's Seekers of the Sky duology, Junior Prince Marcus, a bastard son of the Possessor, is hunted by the entire State and, later, by the Russian Khanate as well.
Sort of happens to David in The Bible. He was just a shepherd boy, but was anointed as the king of Israel by the prophet Samuel. Unfortunately, Saul was still the king, and was not very excited about this. David eventually ends up on the run from Saul.
Odysseus sort of runs into this situation; though he isn't directly ousted, his prolonged absence causes his realm to be taken over and plundered by his wife's villainous suitors, and when he finally returns from his various voyages, to reclaim his land he needs to disguise himself as a beggar, gather his allies, and fight a battle against a small army of noblemen.
Robin Hood, in those stories that make him the Earl of Huntingdon or Robert/Robin of Locksley.
Windswept Questant/White Queen of Homestuck. She driven away from Prospit after Jack Noir attacks and becomes an Exile.
Snowman is also later revealed to be the former Black Queen of the Trolls' session, and joined The Felt after Spades Slick (along with the help of the Troll players) ousted her during the game.
Prince Alfon in Birthright was sent to another country after his homeland of Shartoa was invaded by the Tosiu. Ten years later he returned to start a populist revolution.
Many nobles in the Fire Emblem series become disenfranchised and hunted before either joining or forming a massive army to reclaim their kingdoms, starting from the original Marth, and later reflected by supporting Characters in the Tellius universe in Elincia of Crimea and later Sanaki of Begnion.
Gorath from Betrayal At Krondor, who turns out to have been Delekhan's rival for the throne (though he refused to compete for it) before his exile.
In Dragon Age: Origins, the Player Character of the human noble origin goes rogue. Subverted in that it is not because he/she is a Cousland but because he/she is a Grey Warden.
In the prequel novel The Stolen Throne, Moira Theirin the Rebel Queen is the rightful ruler of Ferelden, whose father was forcibly removed from his throne by an Orlesian usurper. Since then, she has formed a rebellion against the occupation. After she is assassinated, her young son Maric becomes the King-in-exile. There are also several nobles that join the rebellion.
In Fable 3 the main character is pretty much this right from the get-go.
Ramza Beoulve of Final Fantasy Tactics was a member of the noble Beoulve family; as a knight turned mercenary, he eventually defected upon discovering that his brother killed his father and started to orchestrate a rebellion to gain the throne. For the remainder of the game, he remains a fugitive, trying to avoid capture by the kingdom and the Church.
Princess Sadira of Vanguard Bandits becomes a fugitive of her own Empire after Faulkner usurps it.
White Russian exiles from the Russian Civil War played a part in politics in the 1930s and 1940's. Many were merely mercenaries or spies, but the most notable of them was Field Marshal Mannerheim. While not all of these were hunted some were, or had reason to fear that they were.
Almost every European head of state in World War II.
Robert the Bruce.
Harald Hadrada. He had to flee Norway after his brother, the king, was killed in battle. After some time in Russia, he worked as a mercenary for the Byzantine Emperor, soon becoming a commander. After learning that his nephew had taken control of the country, he used his newfound wealth to raise an army and force his nephew to accept him as co-ruler. A couple years later, the nephew died, leaving him sole ruler.
