When putting together the political structure of a fictional country, the default setting for fantasy is a kingdom, for obvious reasons. People know, more or less, what a kingdom is and how it works. But history does furnish us with some other interesting forms of government/inheritance that could be of use in fantastical writing.
One of the most unusual, which I read about in Vanished Kingdoms by Norman Davies, was the system used by Montenegro prior to the end of World War One (when it got shafted by Serbia and was betrayed by its allies, sadly including the UK). The country was ruled by a prince-bishop. Not only that, the nephew of the ruler was the chap who inherited. That sounds very odd, but it does have an advantage. The ruler’s brother will not seek to claim the throne, because his son will inherit. The ruler’s son will not, but his son might. The zig-zag, as it were, inheritance could actually be more stable than the more obvious eldest son model used in traditional monarchy.
Rome’s emperor was often the son of his predecessor, but this was never the case during the Golden Age of Imperial Rome. From Nerva to Marcus Aurelius (who buggered it up by letting his psychopathic son Commodus get the job, after Lucius Verus) the emperor adopted an heir. It worked rather well. This period saw an end to the civil wars that had preceded and would succeed it, and the extent of the empire grew to its largest size.
Reincarnation is used to determine the Dalai Lama. It can take a few years for the successor to a departed Dalai Lama to be found. The People’s Republic of China (which is in control of Tibet, of course) has stated that it has supreme authority on the selection of the next Dalai Lama, which may make things rather messy.
A slight twist on monarchy is the diarchy (two kings). In a small way, England had this in 1689 when King William and Queen Anne were both monarchs in their own right. However, Sparta had two kings as a matter of course. Naturally, there’s scope for regal rivalry, but it also enables supreme authority to be in two places at once (as per the two consuls of Rome).
My knowledge of Renaissance Venice is not fantastic, but the power of the Doge and its shadowy council of wealthy chaps [not its official title] was significant. The Doge Enrico Dandolo was instrumental in the Fourth Crusade attacking Byzantium, which had the short term impact of improving Venice’s power significantly, and the long term impact of bringing down the Eastern Roman Empire and allowing the Ottomans to overrun half of Europe. The Doge was elected but usually served for life, although his executive power was diminished later on. Wealthy individuals formed a sort of aristocracy within Venice, and they had the real power for much of the republic’s life (it ended with Napoleon’s conquest).
The Ghibellines and the Guelfs [which sound a shade Red Dwarf] were the supporters of the Holy Roman Emperor and the Pope, respectively. Centuries ago, the Pope was not merely a spiritual leader, but one who also held temporal power. For a time, the Pope also had a dominant moral position, approving (or not) marriages and playing a critical diplomatic role to foster peace (or not) in Europe. The Pope was, as now, chosen by the college of cardinals.
So, there we are. A mixture of gaining power through election, inheritance, adoption and reincarnation, and roles that mingle the spiritual with the temporal. There’s nothing wrong with kings, but other supreme leadership roles are available.