Royal Highness (His/Her Royal Highness, oral address Your Royal Highness) was a form of address used for members of royalty below monarchs, who were addressed as Majesty.
In the city of Morphoton on Marinus, Ian Chesterton sarcastically refered to Barbara Wright as "Your Royal Highness" and "Your Majesty" as she was tended to by the city's inhabitants. (TV: The Keys of Marinus)
On Peladon, the Third Doctor presented Jo Grant to King Peladon of Peladon as "Her Royal Highness, the Princess Josephine, of TARDIS." (TV: The Curse of Peladon)
The Third Doctor addressed the Draconian prince as "Your Royal Highness". (TV: Frontier in Space)
Prince Reynart, first in line to the throne of Tara, was refered to as "His Royal Highness" by Grendel. (TV: The Androids of Tara)