I like to play with superlatives. I want a government that pretends to be rightly-guided by the most sacred morals and ideals, but in reality is filled with fat cats who only care for themselves. However, the people must not know about it; they know about corruption in other countries, and the clear majority of people truly believe that their country is above that. So how far can I go in my government's corruption?
The general era is something we would recognise as the 15th-16th century. There's steel and gunpowder, but no steam power. There's a literary tradition stretching back a thousand years, but few people are literate. There are religious beliefs but they are not part of the state; in its place is a state philosophy that might as well be a religion. People are steadily growing sceptical of said philosophy, but for now the government (while ostensibly encouraging critical thinking) suppresses diverging thought, passing it off as misguided and/or dangerous.
The government, ruling over a hundred million people of many different cultures in a nation the size of Kazakhstan, works on the basis of the following philosophy: rule is only by the virtuous, and every virtuous person will end up ruling. There's imperial exams taken by every young man and a few bright women (the society is still quite sexist), and those with the best results will end up with roles in the bureaucracy. There's layers upon layers of bureaucrats that report to one another, and at the very top is a council of 52 supreme officers, along with a supreme emperor whose only political power is appointing and dismissing those officers (and giving them supreme moral guidance). The position of supreme emperor is for life, but not hereditary; when one dies, the successor is whoever got the very best score in next year's imperial exams, from a different province in the country (rotating between all the provinces).
Now a lot hinges upon the fairness of these exams, so one of the lowest positions in the bureaucracy is overseeing them. That means that the turnover rate is high among the examiners, so there's little opportunity for corruption to take root there.
Every political position of meaning is part of the bureaucracy; hereditary power transfer is virtually non-existent. Outside the state, power structures exist in the form of religious institutions, and private commercial enterprises. But the church is not quite an authority; the state philosophy is religiously pluralistic and states that all gods live together in one pantheon, leaving open which one is the head god (if any). And this is long before corporations became the lobbying influence we have today; a few figures like Jakob Fugger exist but most businesses are small and family-owned.
That's the theory. Everything stated in the previous section is either accurate or popularly perceived as such. So where can I add corruption? The easiest place is among the bureaucrats. There's so many layers and departments, and people from layer n are generally appointed and dismissed by those from layer n+1. They all socialise with one another so they can get an idea of which of potential candidates for a position would be susceptible to take bribes, and then only promote those people; effectively shutting the true idealists out of the administration beyond a certain layer. Impeachment procedures are rare since people in higher stations are assumed to be more virtuous.
But I want to go beyond that. I am looking for the most effective methods by which the highest officials can evade all scrutiny and turn their backs on the doctrine of virtue deciding position. I want ways to maintain the illusion of social mobility, compatible with the state as described, that allow for large-scale embezzlement and self-enrichment by the powerful.
Take this in a society that is not democratic and without mass media, but also not a totalitarian government that can just make people disappear and it lacks means of communication faster than a horseback courier. How corrupt can I make the country, while everyone still believes that it is not corrupt; and in what ways?
What measures can the government officials and powerful people in general take in this country, to disproportionally enrich themselves, without people at large perceiving their country as corrupt, given the society and history as described? Average citizens who are not closely involved with politics should be able to reasonably think that their leaders are really virtuous and not at all enriching themselves.