Examples of states which benefited from the use of mercenaries on a large scale?

My source here is my general knowledge of Dutch History as well as a course I took on the Spanish Empire. I'm at work, my literature is in another country so I can't really cite anything. I hope someone can give more and better answers to this question, I've just taken one example of an answer to OP's question.

During the renaissance period and the following early-modern period a lot of fighting in Europe was done by mercenaries. The most well known examples are the Italian city states that waged war on each other with mercenary companies most of which were homegrown Italian, although the most famous were Swiss and the most numerous in Europe German (which was overpopulated at the time).

The state that benefitted the most from mercenaries in my opinion is the Netherlands (I am biased as I'm Dutch) or as it was called at the time the Seven United Provinces. Being a decentralized brand-new republic which heavily focussed on cities instead of nobility(which is the traditional basis of armies under the fuedal structure which was still common in Europe at the time) it was impossible to recruit an army big enough to defeat the Spanish who we had risen up to in rebellion. There were homegrown military units like 'schutterijen' a form of militia too, but they were mainly to defend the cities or areas they came from.

Just some facts here before I continue on the mercenaries to paint a picture of the scale and size of the use of mercenaries in the Netherlands. The Dutch Rebellion as it is known internationally or the '80 Years War' as it is known in the Netherlands lasted, you guessed it, about 80 years with some lulls in the fighting and even a 12 year cease-fire. Nonetheless during most of this time the Spanish tried to subdue the Netherlands in quite a brutal fashion. The largest army the Spanish had during this time was always the Army of the Low countries and I took a course on the Spanish Empire once were it was said that the extended military expenditure in the Low Countries was what got the Spanish Habsburg Empire to it's knees. Basically most of the gold and silver coming from the american colonies went straight to the armies in the Low Countries, when the gold didn't arrive the Habsburgs had to declare bankruptcy. Furthermore the crucial importance of the Low Countries can be seen in Geopolitical moves made by the Habsburgs, for example the wars fought over Milan with France was not only about control over northern Italy, but also about keeping a safe land-route open to the Low Countries to march the armies through known as the Spanish Corridor (Sail to Genua, Milan, Burgundy, Palatinate (Pfalz) and then the Low Countries) which was fully under Habsburg control.

Anyway, the mercenaries. With no structure to recruit regular troops, a small population yet a shitload of money (the Low Countries were only rivalled by Northern Italy in terms of urbanization, industry and wealth) the obvious choice was hiring mercenaries. Although most of the Dutch strategy was to build fortifications and prevent the Spanish from taking any Dutch cities, armies were necessary to retake lost possessions so they were bought.

The original army that William of Orange marched into the Netherlands in 1568 to ignite the revolution was an army he brought and bought from Germany. Despite the failure of this first campaign it is typical for the rest of the war that the first shots fired in the revolution were not actually by Dutch people(not to mention that William of Orange was German/French himself). I am not going to delve too deeply into the war as it is a whole different subject(interesting but not relevant to your question and I feel like I'm getting carried away.

To get closer to your point, the State Army(staatse leger) was paid for by the republic and generally led by the Prince of Orange who was also Stadthouder of Holland (the Province). My professor at the time estimated that up to 80% of the troops that fought our war were Germans, not only on land but also on water. This was supplemented by actual Dutch people fighting and French Huguenots (calvinists who seeked refuge in the Netherlands). The Dutch fleet, although not reaching it's peak until the Golden Century that directly followed the '80 years war was also mostly manned by Germans. The simple fact was that central Germany was overpopulated, they needed jobs and the Netherlands had the money to pay them to sail, fight and even colonize for them. Without mercenaries the Netherlands would not have existed basically.