I need to dismiss your limitation of no ruling class. No bureaucracy that touches a society can exist without there first being a leadership that acts on the paradox of helping the people for the good of the society. There are always rulers.
Given that one dismissed requirement, let's have some fun. What we want is the most bureaucratically complex drinking bird heat engine we can: something that employs a lot of effort to achieve a balance it can never achieve for no useful purpose at all.1
The Rule of Law
Bureaucracy cannot exist without the rule of law. Therefore, your society must have the structure of law and its attending bureaucracies: the judiciary, the legislature, and the "funding entity" (we'll get to that momentarily).
What's most important about this is that no solution to a problem can exist without first developing a regulation to express the solution. This is the fundamental problem with the rule of law: government has but one "product," law. It can't divest itself, it cannot diversify its interests, it cannot develop new techniques or technologies, because it has but one product: law.2
The Essential Paradox
A world-class bureaucracy cannot exist without one essential paradox: the need to protect the status-quo while permitting ambition. People want to sandbag their economic and social position so that no one can remove it, but want that same defense removed from the next person in line whose position the first person wants. This paradox creates the illusion of an elite class: those who "understand" the system and can take advantage of it. In reality, the elite class are those who learn how to manipulate the system. Understanding it is someone else's problem.
A Problem Needs Solving
No bureaucracy exists simply to exist. They always begin to solve a problem. For example, how do your judiciary and legislature get paid for creating and adjudicating law? Ah... taxes...
And that explains the U.S. Internal Revenue Agency, probably the most powerful and Kafkaesque bureaucracy on the planet.
My point is, you need a problem, and while the problem is ostensibly for the "good of the people" in reality it's not even "good for society" but is only a means to an end, the end represented by the drinking bird heat engine: it's ultimately self-serving.
No interest may reign supreme. If it does, your society must collapse. This requires at least one overriding law (call it a "constitution") that establishes the requirement for government (the rule of society over the individual) while protecting the rights of individuals (the rule of the individual over society). This creates a see-saw effect where laws are continuously created and modified to protect the impossible balance between the two.
Add to this additional competing interests: business, local governments, international alliances, and class structures. I know you're trying to avoid having class structures, but they're impossible to avoid and necessary for a good bureaucracy. The desire to be a little better than your neighbor, a little more privileged, is an unbelievably powerful motivator for increased bureaucracy.
It's Gotta Be Valuable
Whether its taxes or your local Department of Motor Vehicles, the "service" being provided requires the appearance of value. You want access to government contracts, you want to drive your car, you want the police to protect you from criminals.... All these things are easily solved by other means (work at McDonalds, take the bus, move to a better neighborhood), but that's how an individual thinks, not how "society" thinks. Put a bunch of us together and we want the shortest path to happiness. That's valuable! So, whatever your bureaucracy does, it needs to be perceived as valuable. Taxes are important! So we need the IRS, right?
You Need Distrust
Finally (I could go on a lot longer, but this answer is getting long...), you need a justification for paperwork — because nothing generates the need for new regulations and laws like the need for paperwork. Paperwork requires instructions, training, processing, disposition, action, and a lot of other things that requires laws (which, in turn, require more paperwork... see?).
And the most obvious way to justify paperwork is to not trust anybody.
Have you ever thought of how to avoid trusting anybody? A needs help from B, who uses paperwork to justify that help which must be validated by C to ensure B isn't giving A a deal they don't deserve (it's the Public Trust, after all!). We can't trust C, so we invite C's peers, D and E to check up on C, who require F to file the paperwork and G to archive it and H to digitize it. And we all know H can't do his job without requesting equipment and supplies via paperwork... which means H = A != B + C + D + E + F + G. Yup, paperwork.
OK, You Need One More Thing: Worry
Finally, finally, you need people to worry about the fact that, because they're in a giant hamster wheel of a bureaucracy, they aren't completely safe. There are performance reviews and layoffs and that ambitious twerp down the hall and the overwhelming desire to not appear less than your neighbor... You need a reason to become the big fish in your small pond... and the only way for everyone to feel more-or-less safe is for there to be a lot of small ponds. Lots and lots and lots.
1 I hereby declare entertainment to be a non-useful purpose. So say we all....
2 I read a book, "metamagical themas," that included an interesting game to exemplify the problem with the rule of law: a group of people gather around a stack of blank slips of paper and a pencil. They can't do ANYTHING without creating an appropriate law through consensus. They can modify laws, but they can't "delete" or throw out laws. Every action requires a law. (We had to develop a law that determined how a player could ask to go to the bathroom, then another to actually go to the bathroom, then another to deal with how to bring said player up to date after he/she returned from the bathroom....) The game gets out of hand very, very quickly.