A bureaucracy was a component of government in which policy was carried out and administered by unelected officials.
In 2286, Doctor Leonard McCoy derided "bureaucratic mentality" as "the only constant in the universe". (Star Trek IV: The Voyage Home)
In 2365, Jean-Luc Picard explained that the Mariposa colony had been lost in the Federation's bureaucracy. (TNG: "Up The Long Ladder")
In 2368, when Dr. Beverly Crusher refused her request to use a genetronic replicator on the grounds that Starfleet Medical had already had rejected her three times, Dr. Toby Russell accused Crusher of hiding "behind the rules of some bureaucracy." (TNG: "Ethics")
In 2370, Constable Odo denounced the Bajoran Provisional Government as "idiotic bureaucracy" following as its decision to recall Major Kira Nerys to Bajor. (DS9: "The Circle")
Discussing why she stayed aboard the USS Discovery for its one-way trip to the 32nd century, Philippa Georgiou opined to Commander Nhan that Section 31 would have asked her to lead them eventually, "and bureaucracy is where fun goes to die." (DIS: "Far From Home")