As many as he or she wants. But we’re particularly impressed with Juan Carlos I of Spain, who claims no less than 38 designations on his royal business card:
King of Spain, of Castile, of León, of Aragon, of the two Sicilies, of Jerusalem, of Navarre, of Granada, of Seville, of Toledo, of Valencia, of Galicia, of Sardinia, of Córdoba, of Corsica, of Murcia, of Jaén, of the Algarves, of Algeciras, of Gibraltar, of the Canary Islands, of the East and West indies, of the islands and mainland of the Ocean Sea; Archduke of Austria; Duke of Burgundy, of Brabant, of Milan, of Athens and Neopatria; Count of Habsburg, of Flanders, of Tyrol, of Roussillon, and of Barcelona; Lord of Biscay and of Molina de Aragón; Captain General and Supreme Commander of the Royal Armed Forces; Sovereign Grand Master of the Order of the Golden Fleece and of the Orders Awarded by the Spanish State.
By comparison, the British look positively restrained. Here’s the queen’s official title:
Elizabeth the Second, by the Grace of God, of the United Kingdom of Great Britain and Northern Ireland, and of her other realms and territories, Queen, Head of the Commonwealth, Defender of the Faith.