So it depends on if you're using the titles duke, king, and emperor in a European sense and only in that sense, or if you're also inclusive of other titles around the world which are loosely translated as duke/king/emperor.
For example, in the East Asian cultural sphere, Chinese people only recognised the title of emperor (huangdi) when it applied to the Emperor of China. There was only one emperor and that was always the emperor of China, which included when it was ruled by the foreign dynasties of Yuan and Qing. Under this system, the differences between the three titles you used would have been that there was one emperor, the title of king referred to those who ruled realms non-Chinese, and dukes were under the rule of the Chinese emperor as a basic explanation. Importantly, this did not mean that these rulers all used those titles domestically. The King of Japan was called the emperor of Japan only in Japan; while the titles changed over the years, the Chinese version of the title was generally "King of Japan" as was the Korean version of the title.
So to answer your question, nothing stops an independent ruler from simply claiming the highest possible title for themselves. The question is: what is the point of it? Sometimes, there is value. For example, Korea declared itself an empire following the first SJ war, without any territorial change or new conquest. However, this was of symbolic importance as it put Korea, in the East Asian sphere, as symbolically on the same level of China and as a fully independent entity rather than a nominal vassal of China. At other times, you could cause great offence to neighbouring countries or create implications that you might not entirely be comfortable with. In the case of China, to claim an imperial title in your correspondances with China was to actively challenge the established "world order" in Chinese eyes.
I'm admittedly not familiar with European title history but what I do know is that in Western Europe, the only "empire" that was legitimate was the Holy Roman Empire, which inherited its titular claim from the Roman Empire itself (which is why Russia, which claimed to succeed the Eastern Roman Empire, claimed the title of emperor as well). During the period post-Roman collapse till the Enlightenment, you see that the HRE is the sole entity in Western Europe which claims the title of emperor. While the title of emperor itself could be contested, the legitimacy of it as the only imperial position was not. Names such as the "Spanish Empire," or "Portuguese empire," or "Angevin empire," (I could go on and on), etc. are all anachronistic and were typically ruled by Kings or not truly unified polities (i.e. the Angevin empire). I'd look to someone else for more explanation on the European tradition with regards to this but hopefully this was informative.
Edit: seeing /u/foamsquad 's answer reminded me of something to add.
(Based on your inclusion of "duke" I am assuming you are focusing your question in Western history)
The major difference between the three is the amount of territory they have under their control, and whether or not they control that territory in someone else's name (not including God). A duke is very high up in the "chain of command" as we could call it, but does not have the status of king. A duke could have vassals under his command controlling lands in the duke's name, while the duke himself would often (but not always) be holding lands in the name of a king. There are exceptions to this allegiance requirement such as can be seen in the Italian city states, where dukes often reigned over small pieces of territory with no king above them. The land that such a duke reigns over is referred to as a duchy, while the king's is a kingdom (see a pattern?). The difference between such a duke and a king comes down to territory and recognition. The tier between a duchy and a kingdom is a principality, which is ruled by a prince (not to be confused with the royal title granted to the legitimate male offspring of a king). A principality is also historically a common type of state in the Italian peninsula.
A king is the sovereign of a state, and may have dukes as vassals (as well as other ranks of individuals such as counts, barons, knights etc.) while a duke would not have other dukes subservient to him without an alliance between two states formed. A king's peers are only other kings, and his superior can only be found in the papacy or an emperor, keeping in mind that some later kings would discard ties to the Vatican and achieve a truer form of sovereignty.
An emperor is trickier because the requirements change over time, as most things do. In pre-eighteenth century Europe to be emperor meant to be the Holy Roman Emperor, which was a title that was granted by the Papacy and not one that was inherited. To best describe an emperor, it is best to acknowledge the other half of your question: What stops any independent ruler from simply claiming the highest possible title for himself ? To become an emperor, one cannot simply declare themselves to possess the title while a single kingdom is all they control. It would cost far too much face and would be a challenge to the status of other kings, assuming the individual in question was a king and not a prince or duke, which would be even more problematic. Asserting oneself as emperor implies that multiple kingdoms are under the control of that single individual, which if it simply were not the case then our new (and false) "emperor" would be laughed at by proper monarchs who knew their own status.
Conversely, not all who reigned over multiple kingdoms assumed the title of Emperor. This assumption of rank changed from state to state. Philip II of Spain for example was certainly in charge of an empire - one of the largest in human history, and was even the son of Charles V, a Holy Roman Emperor, yet he did not claim the title of Emperor for himself. Catherine II of Russia however did claim the title of emperor (Empress in the female rendering) over the Russian Empire. Later in history as the Age of Exploration concluded almost every major European power would at some point be an empire, including Britain, France, Germany, and Russia. These countries all held territories outside of Europe, most notably in Africa and Asia, while claims to territories in the Americas were gradually fading for reasons varying enough to warrant their own threads. But again, things change over time, and because human society was shifting away from monarchies and kings and queens were losing power, monarchs in these states were not always named emperor because they were not in charge of the empire. In the British case we know that Parliament was slowly and steadily gaining power while the monarchs lost power. This is why during World War II, despite the British Empire retraining territories outside of Britain's traditional borders, King George was King George, and not George, Emperor of Britain. Conversely, in the late nineteenth century William I became the head of the German state, which was an empire, and therefor assumed the title of Emperor of Germany.
The major distinction seems to be that if the monarch is the true head of state then they could claim the title of emperor, but cases such as Philip II means that the rules behind who considers themselves emperor or not changed over time. Catherine II and Philip II lived in very different times and different places, and the good historian will always account for even small differences let alone such massive ones.
For TL:DR people: you can't just say you are above your own station because your peers will keep you in check. If you want to call yourself king you had better have dukes under your command who will back you up. If you take over other kingdoms you can call yourself emperor, but some people historically did not. Thank you for coming to my TED Talk.
Sources/Suggested Reading on the Topic:
- Mary Fulbrook's A Concise History of Germany
- Christopher Wickham's *The Inheritance of Rome (*Discusses how Europe adapted to the collapse of the Roman empire up to 1000 CE)
- [Edited Volume] Early Modern Spain: A Documentary History - This contains a lot of primary source documents pertaining to the Spanish empire covering a large amount of topics. My favorites are the "debates" of Juan Ginés de Sepúlveda and Bartolomé de las Casas.