Hereditary Republic

Most countries today are republics, wherein the leaders are elected (or at least pretend to be).

There are, however, some republics in which the power resides in the hands of a single family, just as it would in a monarchy, except they refer to their leaders by republican titles (usually president), and there is no actual law stating that the succession works thusly.

Not quite the same as the Peoples Republic of Tyranny; that's when the country doesn't seem to fit the "democratic" or "people's" descriptor. Hereditary Republic is when it's the "Republic" part that's in doubt. May even have a President for Life in charge.

Inverse of Elective Monarchy

Literature

 * The Honor Harrington books feature a few of these (of course):
 * The People's Republic of Haven had a sort of nobility in the form of the Legislaturalists, the powerful families that made up the constantly-elected leadership of the nation. Their leader was Hereditary President Harris until he and most of the rest of the Legislaturalists were assassinated as part of a coup by what would become the Committee of Public Safety.
 * The Republic of Monica, which features in the later books, has a similar form of government, though they maintained that their their leader, President Tyler, had been legitimately elected for every consecutive term he served. Just like his father and grandfather had. And just like his son would.
 * In Foundation, The Republic of Korell.

Live Action Television

 * The Centauri Republic on Babylon Five was very much this trope.

Tabletop RPG

 * Classic Traveller supplement The Traveller Adventure, adventure "The Wolf at the Door". On the planet Aramanx the Republic of Lanax has three co-equal heads of state. Originally they were selected by a democratically elected Administrative Council, but after a "political reorganization" at least two of the three positions are always held by members of the Klaven family, which gives them control of the country.
 * Shadowrun supplement Tir na Nog. The government of the title country (which used to be called Ireland) appears to be democratic, but is actually under the control of the Danann Families.

Real Life
Hoo, boy...


 * North Korea. The first General Secretary of the Workers' Party of Korea (which made him the leader) was Kim Il-sung. The next was Kim Jong-il, his son. After his death, his son Kim Jong-un, was announced as the successor. Oh, and Kim Il-sung is still president, years after his death.
 * The Republic of Nicaragua, under the Somoza family.
 * The Roman Republic ended up like this.
 * England (and Wales), Scotland and Ireland were a republic under Oliver Cromwell, who was succeeded by his son Richard, though this was mainly because Cromwell most emphatically refused the crown that Parliament was fully prepared to offer him. Other than that he was the King in everything but name.
 * Haiti's François Duvalier was succeeded by his son Jean-Claude Duvalier
 * Syria, where Hafez al-Assad handed off power to his son Bashar.
 * Averted by Egypt, where Hosni Mubarak might well have left the presidency to his younger son Gamal had it not been for the Arab Spring. Indeed, trying to avert this was one of the main reasons Egyptians revolted in the first place--although it's likely that there would have been a revolution anyway even if Mubarak had promised not to give Gamal the presidency. You see, the hereditary succession was seen more as a symbol of the regime's corruption, and while most Egyptians were opposed to the idea on principle, most would also admit that they wouldn't have had much of a problem with it if it didn't occur in the context of a corrupt, authoritarian, and cynical regime.
 * Raul succeeded his brother Fidel Castro as President of Cuba. Subverted in that Raul was a leading political figure in his own right and that none of the politicians tipped as likely successors are related to the Castro brothers.
 * Azerbaijan. The previous president, Heydar Aliyev, made his son Ilkham the next president.
 * The United States has had several political dynasty families (generally at state or local levels of government) with associated political machines (and sometimes with high levels of corruption and patronage). These include the Daleys of Chicago, the Byrds of Virginia, the Kennedys of Massachusetts, or the Tafts of Ohio.
 * When the Democrats were choosing a candidate for the 2008 election, it was noted that if Hillary Clinton became president, and served two terms, the USA would have been ruled for 28 years by members of two families.
 * Then subverted when she wasn't nominated. Though she was later appointed Secretary of State.
 * There has been three cases when a president has been the descendant of a previous one: John Quincy Adams and George W Bush were the sons of John Adams and George HW Bush, respectively, while Benjamin Harrison was the grandson of William Henry Harrison. None, however, directly succeeded their ancestors (Dubya came closest--he entered office only eight years after his father left it.)