Countries without a president or king
Firstly it's important to point out that all countries have a 'head of state' and a 'head of government'. In some countries these are two separate posts, and in other countries they are combined into one single post.
For example in India the President is the head of state, but the Prime Minister is the head of government. Conversely, the President of the United States is both the head of state and head of government at the same time.
In some countries, depending on whether the president is both the head of state and government can affect their roles, responsibilities, and powers.
In the case of monarchies, some kingdoms have a king/queen who is head of state in a largely ceremonial role with limited or no real powers to govern, and they have a prime minister (or similar title) as head of government. In other cases the king/queen is both head of state and government.
Most countries in the world have a 'president' as their head of state. As of writing this post there are
146 countries with presidents as the head of state;
96 of them have a president who is head of state,
50 of them have a president who is head of state and government.
There are also several countries with a king/queen as their head of state.
There are currently 15 countries with a king:
Bahrain, Belgium, Bhutan, Cambodia, Eswatini, Jordan, Lesotho, Morocco, Netherlands, Norway, Saudi Arabia, Spain, Sweden, Thailand, and Tonga.
There are also 17 countries with a queen: one is Denmark. The other 16 are all members of the Commonwealth Realms, which Elizabeth II is leader, and therefore head of state of all member countries which includes; Antigua and Barbuda, Australia, Bahamas, Barbados, Belize, Canada, Grenada, Jamaica, New Zealand, Papua New Guinea, Saint Kitts and Nevis, Saint Lucia, Saint Vincent and the Grenadines, Solomon Islands, Tuvalu, and of course United Kingdom.
Now we get onto the interesting examples:
Japan - Emperor
There have been many countries led by emperors throughout history, such as China and Ethiopia, but Japan is the only empire left in the world today. Japan is like many other constitutional monarchies where the emperor once governed the country directly but over time and now serves as a purely symbolic/ceremonial position with no actual power to govern.
Luxembourg - Grand Duke
Luxembourg is another example similar to Japan. Instead of an emperor it has a Grand Duke, which is a ceremonial position. There have been other grand duchies in the past such as Finland, Lithuania, and Tuscany. Luxembourg became a grand duchy in 1815, and is the only grand duchy left in the world today.
Liechtenstein and Monaco - Prince
Liechtenstein and Monaco are both tiny microstates in Europe. They are monarchies, but instead of a king they have a 'prince'. These tiny countries are remnants of a time where there were once many principalities, which generally were allied with or subservient to larger kingdoms. There are two ceremonial principalities as well; Asturias in Spain, and Wales in the UK.
Kuwait and Qatar - Emir
The leader of an 'emirate' is called an emir, an Arabic word used to refer to a king or aristocratic noble. There are many emirs in the Middle-East, such as in Saudi Arabia. The United Arab Emirates is a federation of seven different emirates, but the head of state uses the title president, and functions as a sort of high-king of the emirates. There have been other emirates of Arabic speaking states, but Qatar and Kuwait are the only independent states left ruled by an emir.
Brunei and Oman - Sultan
Similarly to emir the word sultan derives from Arabic, and is a title that has been used by numerous Muslim states in the past. It derives from an Arabic word meaning 'strength' or 'authority'. Sultan is similar to king, but more specifically has a religious significance as well as political. Some countries have changed the title of their leader from sultan to simply king, to emphasise a more secular nature to the role such as Morocco. Today there are many sultans in Malaysia who rule sub-divisional states, but the only two countries left that are ruled by sultans are Oman and Brunei.
In practice, whether they hold the title emperor, grand duke, prince, emir or sultan, they basically fulfil the same function as a king/queen, whether as a constitutional monarch or an absolute monarch, the title they use is simply different due to historical circumstances. Naturally the title of a king/queen varies depending on the language of their country anyway.
In Norway the king is known as konge
In Spain it is rey
In Arabic speaking kingdoms the king is malik
The king of Tonga is called tu'i
The king of Eswatini is ngwenyama, translating as 'Lion'
The king of Bhutan is the Druk Gyalpo meaning 'Dragon King'
But there are still some countries whose leader is not simply a king with a different name, they have an entirely different setup altogether.
San Marino - Captain(s) Regent
San Marino is quite special because its head of state is called 'capitani reggenti', or captain regent in English. And there is not just one, but two captains regent. This is a rare instance of a country having two leaders instead of just one, and derives from the custom of the ancient Roman Republic, which was ruled by two consuls. In fact the captains regent were originally called consuls when the roles were founded in the early 13th century. The names and roles of the title changed gradually over time. Today the role of the captains regent is mainly symbolic.
The captains regent must be from San Marino, and are elected every six months (they are changed every 1 April and 1 October). You can be captain regent more than once, but not within the space of three years. In 1972 women were allowed to become captain regent. Since then, through this unique system, San Marino has technically had more female heads of state than any other country, 17 in total so far.
Andorra - Co-Prince(s)
Like San Marino, the tiny nation of Andorra has two heads of state, each called a co-prince. Unlike San Marino, Andorra's co-princes come from outside the country. One comes from Spain, the other comes from France. These are not princes like for Liechtenstein and Monaco, the co-princes are appointed rather than hereditary positions. But this system derives from the ancient tradition of Andorra once being controlled jointly as a condominium by powers from either side of its territory.
Nowadays, whomever is appointed the bishop of the Catholic diocese of Urgell on the Spanish side of the border becomes one co-prince, and whomever is elected president of France becomes the other co-prince. They hold the titles of co-prince ex officio to their primary title.
The powers of the co-princes are largely symbolic. Andorra has a parliament led by a prime minister actually from within Andorra. Andorra is one of only a few countries with more than one head of state, and the only country with two heads of state that don't actually even come from within that country.
Switzerland - Federal Council
The Federal Council of Switzerland is a body of seven people, who collectively serve as the head of state of Switzerland. But this council also has a president (and vice-president), who has particular roles and functions within the council. This position rotates every year between the president and the vice-president.
Sudan - Sovereignty Council
Sudan is another special case. Its current head of state is, like Switzerland's, a collective body called the Sovereignty Council, composed of 11 people. Since Sudan's independence in 1956, the Sovereignty Council has ruled Sudan several times in its history. It was reinstated as head of state during a transitional period following the overthrow of the brutal dictator Omar al-Bashir in 2019. The transitional period is scheduled to end in November 2022, where (theoretically) Sudan will be led by a president again.
Bosnia and Herzegovina - Chairman of the Presidency
Another example of a multi-member body jointly serving as head of state, Bosnia and Herzegovina has three presidents all at once. The constitution states that one president must be a Bosniak, one a Croat, and one a Serb. They each take it in turns to be chairman of the presidency.
Vatican City - Sovereign
The pope is elected by the College of Cardinals to lead the Catholic Church, but he also then becomes the head of state of the Vatican City. In this capacity his title is 'Sovereign', which functions in practice as an elected absolute monarchy.
Iran - Supreme Leader
Iran actually has a president, but that is the title of the head of government. The head of state has the much grander title of Supreme Leader, a position which was created after the Islamic revolution of 1979. It is both a religious and political title of supreme authority in Iran. The Supreme Leader has the final say on any government decision, outranks and can overrule the decisions of the president, and appoints the heads of the government and military. The post is for life.
The Supreme Leader is appointed to this post by a group called the Assembly of Experts. In theory the Assembly of Experts hold the Supreme Leader to account and can even dismiss him from his post. The Assembly of Experts are appointed by the 12 member Guardian Council. Members of the Guardian Council are in turn selected by the Supreme Leader.
There have so far only been two Supreme Leaders of Iran. In effect the Supreme Leader is an elected monarch.
Malaysia - Yang di-Pertuan Agong
The head of state of Malaysia is called Yang di Pertuan Agong, which means "He Who is Made Lord". Unlike many other monarchies, Malaysia is a federal elected monarchy. There are several states in Malaysia, nine of which are led by hereditary rulers (usually called sultans). Any of these are eligible to become Yang di-Pertuan Agong, which in a sense acts like a high-king, similar to the president of the United Arab Emirates. One of them is elected to the post by the others. The term lasts five years, and in theory one can serve more than one non-consecutive term.
Cambodia is also an elected monarchy, but functions differently to Malaysia. The king is chosen from any of the male descendants of the incumbent king and the post is for life.
Samoa - O le Ao o le Malo
If you thought Yang di-Pertuan Agong was a mouthful, then try Samoa's head of state; O le Ao o le Malo, which translates as 'Chieftain of the Government'. Samoa had been traditionally led by tribal chiefs, the Paramount Chiefs of the four main lineages of Samoa called the Tama-a-Aiga. The four paramount chiefs were known as:
When Samoa became independent in 1962 these chiefly titles were essentially incorporated into one title, the O le Ao o le Malo. In practice the function of the O le Ao o le Malo is the same as a ceremonial president, and can serve a maximum of two 5 year terms.
North Korea is a very special case indeed with probably the most complex and bizarre setup of all of the countries in the world. Officially speaking North Korea has two heads of state, both with very long titles:
One is the Chairman of the Presidium of the Supreme People's Assembly, a title currently held by Choe Ryong-hae,
the other is the Chairman of the State Affairs Commission of the Democratic People's Republic of Korea, also known more succinctly as the Supreme Leader, which is held by Kim Jong-Un.
But North Korea does in fact have a president: Kim Sŏng-ju, better known as Kim Il-Sung, the founder of North Korea, was declared Eternal President of the Republic in 1998, a title he still holds, even though he died in 1994.