Every philosopher has a foundation on which they build their theory: Plato had "the Forms," Hegel had "Spirit," and Derrida had… okay, we still aren't entirely sure what Derrida's theory is, much less what he built it on, but the point is that these guys are all about power.
Whether it's about dominating other people, dominating oneself, or just plain dominating the conversation, these guys see power as the motivating factor behind most—if not all—human behavior. Though Michel Foucault took over as president of the club after beating then-president Nietzsche in an arm-wrestling match over who gets to take the club's gavel home, Nietzsche maintains an active membership and has awarded himself numerous "Treasurer of the Year" awards in the years since stepping down as president.
Foucault was the first prominent philosopher to call himself a "Nietzschean," but in reality, he had his own thing going on. In what is arguably his most famous work, Discipline and Punish, Foucault argued that the institutions around us, as long as they're backed in some way by state power, are out to control us through discipline and the threat of punishment. It's all a form of massive control, and we're mostly unaware of it.
A real mood-killer, this guy.
In his famous description of life without a state, Thomas Hobbes went down in history as the first thinker to give "power" a prominent theoretical role when he argued that the reason life without a state would be so miserable is because everybody is trying to gain power over everyone else. According to this guy, without some kind of state power, everybody would be killing and stealing all the time.
Though he never went as "mad with power" as his fellow Power Pals, he was the first to make power a central philosophical concern.
Where the founder of psychoanalysis Sigmund Freud saw sex as the root of, well, everything, Alfred Adler saw power dynamics as the root of everything. Don't tell anyone, but he took the notion of the "will to power" right from Nietzsche.