One example of bureaucracy in The Hitchhiker's Guide to the Galaxy is the president himself. In the book, the president wields almost no real power. He is shown to be a figurehead whose primary role is to distract people's attention away from the real source of power. This is very unlike the role that the president of the United States has, though perhaps much more similar to the current English monarchy. Actual political power has been taken away from the royal family, but they do still have quite a bit of sway.
Another example is the Vogons. Here is a quote:
"Here is what to do if you want to get a lift from a Vogon: forget it. They are one of the most unpleasant races in the Galaxy—not actually evil, but bad tempered, bureaucratic, officious and callous. They wouldn't even lift a finger to save their own grandmothers from the Ravenous Bugblatter Beast of Traal without orders signed in triplicate, sent in, sent back, queried, lost, found, subjected to public inquiry, lost again, and finally buried in soft peat and recycled as firelighters."
In other words, the Vogons are stuck in bureaucratic redundancy. They aren't so much concerned with right or wrong, good or bad. They aren't trying to be mean or nice. They are simply following the letter of the law and making sure that all proper and correct procedures are followed for everything. What should be an easy quick decision about anything takes the Vogons an eternity because they have to check it, double check it, submit the proper forms, check for authenticity, and then get proper authorization for the forms, etc. It is a bureaucratic nightmare.