The earliest example would be the Delian League of city-states c.477 B.C., making this one Older Than Feudalism. The League was even the Trope Namer, since the position of leadership within the league was referred to as "hegemon". This hegemon, to nobody's surprise, was Athens, to the point where the League was often called the Athenian Empire.
The Romans were masters at this, and many client states/vassals had already been so 'romanized' that when the Romans actually moved in to annex them, the people didn't notice (or realize they were, until then, technically still independent).
They were so good that the one time it backfired, which resulted in the Social War (as Rome's federates were called socii in Latin), it was because they refused to annex their Italian vassals: the socii had the duty to provide half the legions for any given campaign, but obtained little of the profits and had no say in the external politics, and when the Senate redistributed public land while refusing to grant them Roman citizenship they finally revolted. The Romans managed to win militarily, but quickly passed a law that made all the loyalist Italian communities into Romans and another that allowed people from the excluded communities to obtain Roman citizenship on an individual basis.
Augustus reputedly wanted and worked hard at making Rome this, thinking of it as a more practical and cheaper option than holding northern Europe by force.
The predilection among many countries today to have imitations of Greco-Roman culture such as "democracy", "senates", and so forth is an example of this. Even the European nations that weren't actually part of the Empire were still affected by it, and that got passed on to their colonies, which became entirely new nations.
Rome's enemy Carthage had their own hegemonic empire in Spain and the African coast from Anfa (modern day Casablanca) to Oea (now Tripoli), and the three main campaigns of their second and most decisive conflict (Italy, Spain and Africa) were effectively attempts at breaking each other's hegemony: Hannibal's Italian campaign had an initial limited success but ultimately failed note There were in fact defections: some Gaulish tribes in Northern Italy, many of the Greek cities in Sicily and Southern Italy, the Samnites, who did not appreciate Roman overlordship, and Capua, formerly Rome's most trusted ally but also the only city who could actually take over the hegemony over Italy. These defections were not sufficient, as the Etruscans, the Latin peoples and the Umbrians remained loyal alongside some Gaulish tribes and Greek cities, and Etruscans, Latins and Umbrians happened to have the largest populations and the richest economies, and their lands surrounded Rome, making it impossible for a direct attack, and were wedged between the Cisalpine Gauls and Samnius, thus keeping the defectors separated and unable to support each other while Rome defeated them one by one., while Rome's Spanish and African campaigns successfully destroyed Carthage's empire.
The United States of America counts as a contemporary example. Its many interventions in Latin America and the Middle East often result in regime changes of supposedly hostile or opposing leaders (particularly those who nationalize their industries); assassinations of rival world leaders; assisted coups that put the armed forces in power; installing of pro-US dictatorships; and establishing puppet states to service American companies. Many countries in the world utilize the dollar, many American corporations have turned multinational, many American products have turned international (ala McDonalds in Japan, of all things), English is the most spoken language in the world, and even NATO is often seen as a mere extension of US military might as opposed to an international coalition.
The United States is often seen as the successor of the English in this regard and differs from the British Empire for the fact that it has so far avoided direct colonization, favoring control over the markets instead. The British Empire initially sought to be a hegemonic empire at the outset but gradually became a military empire upon seeing the competition from other would-be colonial powers (France, Spain, Portugal, Netherlands, Russia, Germany) and the political instability that happened to coincide upon their arrival in a new land. When trouble arose, Britain relied on its vast and powerful Navy, which was essentially unchallenged from the Battle of Trafalgar in 1805 to the Battle of Jutland just over a century later, and unsurpassed until WWII (when it was surpassed... by their allies, the Americans), small, highly trained, efficient and highly mobile armies and its economic influence. The spread of English as the language of power and money (something which can partially be attributed to the US), the retention of Greenwich Mean Time and the resultant system of time zones and the widespread nature of British culture is a testament to this.
The United States was founded with this in mind. After the Revolutionary War, the individual colonies were seen as individual nations that banded together to become an alliance of nations with one entity (The United States Government) being the regulator of all international and inter-state issues. The thought of the United States as a national identity did not gain mainstream support until the fallout of the Civil War (General Robert E. Lee was asked to fight for the Union by Abraham Lincoln the very day Virginia seceded, and Lee sided with his country, which he believed was Virginia; he would have fought for the Union if Lincoln had asked a day earlier). Vestiges of this mentality still exist to this day with the concept of State's Rights (the 10th Amendment to the Constitution explicitly says that if the Constitution does not explicitly reserve a power for itself, than that power is ceded to the State). It's quite common for states to have unique personalities, attitudes, priorities, and interests that are not shared by their neighbors.
Third-world countries that were part of European colonial empires qualified as well. For example, much of The Raj consisted of "princely states" led by native rulers (with titles like nawab, nizam, sultan, raja, maharaja, etc.) who became voluntary vassals of The British Empire in exchange for protection and cementation of their local authority.
The Holy Roman Empire during most of its reign, owing to the difficulty in getting most of the elector princes and other lords to back many serious military campaigns after the Crusades.
The Habsburgs and their family, who were largely the Holy Roman Emperors, from 1440AD onwards are an even better example, holding dozens of countries in their hegemony from the Holy Roman Empire, the Netherlands, much of Italy, Central Europe, and at one point controlled BOTH the Spanish and Portuguese Empires as part of Philip II's Iberian Union.
Vladimir Putin's policy towards the former Soviet republics is essentially creating one of these as a replacement for the USSR. To date it's more or less successful in Central Asia where most of its leaders are happy to support Moscow along with Armenia, Belarus and Moldova. Surprisingly many Balkan and Central European states such as Serbia, Macedonia, Slovakia, Hungary, Bulgaria and Greece are happy to support Russia.
The response from the West has been positively frigid, particularly from the US (which is worried) and Britain (which has had a bone or two to pick with the Kremlin following the Litvenenko assassination). Germany and France, on the other hand, seem to want to avoid another Cold War and are more ambivalent.
Another aspect of Putin's foreign policy is his warm relationship with several right wing populist politicians, some of which are arguably financed with Russian money and/or extolled in Russian media outlets like RT and Sputnik News.
The Achaemenid Empire founded by Cyrus the Great could be considered the Trope Maker. The Empire was highly decentralised and ruled through viceroys called satraps who governed their administrative units, satrapies, in the name of the Persian emperor. Satraps enjoyed almost total freedom to rule their territories as they pleased and were allowed some degree of private armies, their only responsibility towards the crown was the levying of certain taxes and keeping internal peace and banditry down. Satraps would often belong to the native culture of their region, and have Persian advisers to keep a smooth connection between the local rule and the Emperor's court. The Persians also avoided imposing their own religion or culture on subjects, with only the upper class following Zoroastrianism.
Australia, in regards to its immediate neighbors. New Zealand often toes whatever line Australia happens to follow; Papua New Guinea used to be a colony (and is often seen as a puppet) of Australia; Malaysia is too weak to do much; and most refugees in Southeast Asia head over to Australia for better opportunities than in their own countries. Indonesia, the fourth most populated nation and the largest archipelago in the world, is more or less The Rival to Australia, but most of its major parties are corrupt and civil services aren't really helpful to those who don't have the money for it (and there are a lot who don't have the money for it). Its military is roughly on par with Australia and it has a much larger population than Australia, but otherwise, there aren't really any other true opponents to Canberra in its immediate vicinity, so many people (in Canberra, of course) consider Australia to be something of a Rising Empire.
Indonesia itself was one in form of the Majapahit empire, until it fractured in a civil war and rebuilt by new rulers as independent sultanates, until the European colony came...
Even then, the current Indonesia is pretty much Java's sphere of influence because they claimed descent from Majapahit. While officially the country is promoting diversity of its various provinces' culture, the Javanese has so much influence in the governmental, economy, and population ratio that Indonesians often forgot that there's more of Indonesia than the Muslim-majority, Jakarta-dwelling Javanese.
Contrary to popular belief, the Aztecs ran one of these, as an alliance of three city-states demanding tribute, rather than obeisance, from those states they had defeated in battle. They rarely left their own governors in place, and didn't tamper with local religions, relying instead on a combination of cultural hegemony and military superiority to ensure continued compliance with the empire's dictates. That said, a large portion of that tribute comprised Human Sacrifices captured during ritual "Flower wars", which bred enough resentment among their client states that Hernán Cortés was able to recruit many of them as allies. A number of these states, such as the Tlaxcala Confederacy and Cempoala, then became Voluntary Vassals of the Spanish Empire upon the destruction of the Aztecs.
Imperial China operated on this principle for centuries, and is probably the Trope Codifier. In dynastic Chinese political theory, the Emperor was the 'Son of Heaven' and thus the legitimate sovereign of the entire world; even states that weren't directly under Chinese control were expected to recognize this fact. It only really fell apart when the rising power of the industrial West coincided with a period of relative weakness for China; unsurprisingly, the imperialist Western powers (who were used to thinking of themselves as the best people on the planet) weren't too happy to kow-tow to the monarch of some (in their view) backwards nation. A few humiliating (for the Chinese) wars later, it was getting rather difficult to keep pretending that China was the grandest place on earth; this ultimately led to the downfall of last imperial dynasty, and to the founding of Republic of China, which would eventually be conquered (on the mainland) by the People's Republic of China.
And even before that, in China's pre-imperial feudal period, the Zhou Dynasty resembled this for about 600 of its 800 years in power: the king had little real power outside the immediate demesne of the capital, but was nevertheless recognized by the various feudal power-holders as their nominal overlord. During what is often translated as the Period of the Five Hegemons, some lords rose in power sufficiently to informally dominate the other states, without attempting to directly claim the title of overlord. Eventually, though, one of them became powerful enough to conquer the rest of China; this was Qin Shi Huang, the first emperor, and whose short-lived Qin Dynasty is actually the root of our word 'China'.
Some have accused Germany of being this inside the European Union. Whether this is actually true depends a lot on whom you are talking to. The phrase about Germany being "too big for Europe and too small for the world" relates to the problem of Germany being (in theory) able to dominate most European countries economically and (at least in the past) militarily but unable to do really go toe to toe with major empires outside of Europe. Interestingly, some European leaders have actually called for Germany to act more like this (if only to carry the banner of an eventual United Europe), making them in essence voluntary vassals.
While the Mongol Empire is well-known for its military expansion, it made use of its fearsome reputation to make their conquest a lot easier. They spread tales how they burned whole cities to the ground, killed anyone who resisted and enslaved the survivors, but also made a point about sparing anyone who surrendered without a fight. This was preferable since needless brutality would make the resistance a lot more harder to beat. Those who became willing vassals such as the Armenians enjoyed protection by the Mongols and had their religions respected as part of their tolerance policy where Christians, Muslims, Buddhists and other faiths co-existed without persecution or taxation (something unheard of at the time) and in exchange, they'd perform key administrative roles of their empire since their overlords weren't exactly fit for it. They also ensured such safety and security in their lands that a popular saying goes "a virgin carrying a pot of gold on her head could go unmolested across one border of the empire to the other".
The First Mexican Empire spread from what is now Southern US to Panama, however its actual influence in many of this areas was dim to say the least. In Central America for example was almost non-existent and when the Empire fall the Central American province became the Federal Republic of Central America that, again, was more of a loose federal league than anything else. It did not stop what was the Grand Colombia to take away Bocas del Toro to Costa Rica, much to Costa Rica's dismay. Its Constitution based on the American Constitution was basically symbolic and secessionist movements alongside a divisive Civil War that mostly affected Guatemala, Honduras and El Salvador (the centers of power for different reasons) made it all worst. Costa Rica as the most southern part of the Federation was probably the most indiferent to the federal government, it did no took part in the civil war and even declared itself to be separated from the Federation (although it did not declared actual independence) until the civil war was over, basically saying "we would go our way until you put yourself together". The war did was finished, and Costa Rica did re-entered but the damage was done and by 1848 the Federation just collapsed in what are now the republics of Central America. Not that some failed attempts to re-created it (sometimes by military force) haven't existed since.
of city-states c.477 B.C., making this one Older Than Feudalism. The League was even the Trope Namer, since the position of leadership within the league was referred to as "hegemon". This hegemon, to nobody's surprise, was Athens, to the point where the League was often called the Athenian Empire.