Which European country had more kings than any other in the totality of the 1700s?
Does the question mean successive kings of the entire country from 1701 to 1800, or simultaneous kings within the country? Is "country" a modern country or an 18th century country?
Successive kings of the entire country from 1701 to 1800:
Nine Russian Tsars and Emperors.
Ten Bishops of Urgel were co-princes of Andorra.
Simultaneous kings within the country:
Shared rule of the whole country:
Andorra usually had two simultaneous co-princes, but France renounced its share of the joint rule from 1792-1806.
From 24 February 1777 to 25 May 1786 Maria I and her uncle/husband Peter III were joint monarchs of Portugal.
Different kings in different parts of a modern country:
Modern France contains the 18th century kingdoms of France and Navarre that had the same kings in the 18th century. And it contains Corsica that was a kingdom from March to November 1736 and from 1794-1796. And it contains most of the 18th century Kingdom of Arles/Burgundy.
The modern United Kingdom of Great Britain and Northern Ireland contains the 18th century kingdoms of England and Scotland that united to form Great Britain in 1707 and part of the Kingdom of Ireland that united with Great Britain in 1800-1801. The three kingdoms always had the same monarch during the 18th century.
Modern Italy contains the 18th century kingdoms of Italy/Lombardy, Sardinia, Sicily, and Sicily that were ruled by two to three men at a time in the 18th century. Also the city state of Genoa had decreed that the Virgin Mary was its queen during the previous century, if that counts.
In 1700 Castile, Aragon, and Valencia had the same king but different parliaments, etc. During the War of the Spanish Succession the French claimant Philip V consolidated them into a unified kingdom.
18th century realms with more than one king at a time in different territories:
The Holy Roman Empire contained the kingdoms of Germany and Italy/Lombardy whose kings were the same person as the Emperor since 972, and the kingdom of Arles/Burgundy whose kings were the same person as the Emperor since 1032. Also the city state of Genoa had decreed that the Virgin Mary was its queen during the previous century, if that counts.
It also contained Bohemia that was a hereditary and/or elective kingdom since 1198. By various historical circumstances the hereditary King of Bohemia and the elected Emperor were usually the same person during the 18th century.
From 1741-1743 there were two rival monarchs of Bohemia, Queen Maria Theresa and Emperor Charles VII, Elector of Bavaria.
In the 18th century there were two kings of the Romans and future emperors crowned during the lifetimes of reigning emperors. Future Emperor Joseph I from 23 January 1690 to 5 May 1705 during the reign of Emperor Leopold I and future Emperor Joseph II from 27 March 1764 to 18 August 1765 during the reign of Francis I.
How many fiefs and principalities in any one 18th century realm were ruled by rulers of another realm during the 18th Century?
1) The Duchies of Milan and Brabant, etc. in the Holy Roman Empire were ruled by King Charles II of the Spanish kingdoms until he died on 1 November 1700. Then various Spanish kingdoms and duchies, etc. were in the possession of the two rival claimants until the War of the Spanish Succession was settled by the final peace treaties in 1714.
Note: the king of the Spanish Kingdoms in Europe and the New World was also the Duke of Milan, Brabant, Limburg, Luxemburg, etc., etc. in the Holy Roman Empire from 1516 to 1700. And also the king of Portugal from 1580 to 1640.
2) Kingdom of Sicily Ruled by King Charles II of the Spains, Duke of Milan, brabant, etc. in the Holy Roman Empire, until he died on 1 November 1700. After the War of the Spanish Succession possessed by the Duke of Savoy, etc. in the Holy Roman Empire until 1720. Possessed by the King of Bohemia, Archduke of Austria, Duke of Milan & Brabant, etc. in the Holy Roman Empire (Emperor Charles VI) 1720-1735.
3) The other Kingdom of Sicily. Ruled by King Charles II of the Spains, Duke of Milan, Brabant, etc. in the Holy Roman Empire, until he died on 1 November 1700. After the War of the Spanish Succession possessed by the King of Bohemia, Archduke of Austria, Duke of Milan & Brabant, etc. in the Holy Roman Empire (Emperor Charles VI) 1714-1735.
4) Kingdom of Sardinia. Ruled by King Charles II of the Spains, Duke of Milan, Brabant, etc. in the Holy Roman Empire, until he died on 1 November 1700. After the War of the Spanish Succession possessed by the King of Bohemia, Archduke of Austria, Duke of Milan & Brabant, etc. in the Holy Roman Empire (Emperor Charles VI) 1714-1720. Possessed by the Duke of Savoy, etc. in the Holy Roman Empire from 1720.
5) The Kings of France ruled various fiefs in the Kingdom of Arles or Burgundy in the Holy Roman Empire.
In 1789 Louis XVI as Count of Provence and Forcalquier used the title:
Louis, par la grace de Dieu, roy de France et de Navarre, comte de Provence, Forcalquier et terres adjacentes
in documents in Provence and Forcalquier, thus acknowledging Provence and Forcalquier as possessions in a different country outside of France. http://eurulers.altervista.org/
6) In 1747 William IV (1711-1751), Prince of Nassau-Dietz in the Holy Roman Empire, titular Prince of Orange, became the first hereditary Stadtholder of the United Netherlands, a position roughly equivalent to hereditary president or to king.
7) In 1714 George, Elector and Duke of Brunswick-Luneburg (Hanover) in the Holy Roman Empire, became King George I of Great Britain and Ireland, etc. And Great Britain and Hanover remained united until the end of the 18th century.
8) The County (Duchy since 1474) of Holstein in the Holy Roman Empire and the Duchy of Schleswig in the Kingdom of Denmark became more or less united with each other and the Kingdom of Denmark in 1460 when King Christian I of Denmark, Norway, and Sweden inherited them.
Schleswig and Holstein became divided among various heirs but the King of Denmark and Norway always ruled part of Holstein until 1773 when Christian VII became ruler of all Holstein. The king of Denmark and Norway ruled all Holstein for the rest of the 18th Century.
From 1667 to 1773 the king of Denmark and Norway was also the Count of Oldenburg in the Holy Roman Empire.
9) Sweden acquired several German principalities and fiefs in the 30 Years War and the King of Sweden and Grand Duke of Finland remained a prince of the Holy Roman Empire though possession of all or part of Pomerania from 1648 to 1806.
10) In 1701 the Elector of Brandenburg, Duke of Kleve, Julich, Berg, etc. in the Holy Roman Empire, Duke of Prussia, etc., became King in Prussia, a situation lasting until 1806.
11) Frederick Augustus I The Strong (1670-1633), Elector of Saxony in the Holy Roman Empire, became King Augustus II of Poland and Grand Duke of Lithuania from 1697 to 1704/06 and from 1709 to 1733. His son Augustus Elector of Saxony became King of Poland and Grand Duke of Lithuania from 1734 to 1763.
12) The Archdukes of Austria, etc. in the Holy Roman Empire were always kings of Hungary and Croatia during the 18th century. The exact title was: "King of Hungary, Dalmatia, Croatia, Slavonia, Rama, Serbia, Galicia, Lodomeria, Cumania, and Bulgaria".
13) Charles Peter Ulrich (1728-1762) Duke of Holstein-Gottorp in the Holy Roman Empire, was briefly proclaimed King of Finland in 1742. He became Emperor Peter III of Russia from 6 January to 9 July 1762. He was deposed by his wife who became Empress Catherine II.
Their son Paul (1754-1801) - Emperor Paul I from 1796 to 1801 - remained Duke of Holstein-Gottorp until 1773 when Paul traded Holstein-Gottorp to the King of Denmark and Norway for the County of Oldenburg. Paul then gave Oldenburg to his cousin Frederick Augustus I.
14) The city state of Jever in the Holy Roman Empire eventually was acquired by the Princes of Anhalt-Zerbst. Empress Catherine II of Russia was a princess of Anhalt-Zerbst and when her brother died in 1793 Jever passed to Empress Catherine and her successors until Napoleon seized it in 1807.
So during the reign of Peter III of Russia from 6 January to 9 July 1762 there were ten foreign rulers who possessed fiefs in the Holy Roman Empire (or fief holders in the holy Roman Empire who ruled foreign realms): The Stadtholder of the Netherlands, the Kings of Sardinia, France & Navarre, Great Britain & Ireland, Denmark & Norway, Sweden, Prussia, Poland, Hungary & Croatia, etc., and the Emperor of Russia.
If this question means ALL PARTS of what is now one European nation, the answer is:
Germany. Since Germany had by the 18th century fractured into dozens of tiny, independent states, and each little kingdom had its own ruler, Germany in the 1700s had a phenomenal number of minor kings, dukes, and other monarchs. For more see : Wikipedia—"Germany in the Eighteen Century."