I'm intentionally posting before reading the rest of the thread, so sorry if these are already covered. I'll catch up in a bit.
Objectivism
Though I'm loathe to label this a philosophy, that label is often misapplied by its adherents, and it seems to be the dominant philosophy in most D&D games in my experience. The characters tend to only care about themselves, don't care about anything in the world around them unless it can be somehow turned to their own personal benefit, and there's a strong current of that old time might makes right, social darwinism, and fascism. The justness of a cause is determined by who wins any given fight, strength is its own reward, and anyone not strong enough to maintain whatever it is they have doesn't deserve to keep it. To me, this is where murderhobos live. Even the troupes who try to do good are still mostly engaged in selfish adventuring, seeking rewards rather than doing what's right for its own sake. You can see this seep into the community at large with arguments about alignment, a push for moral relativism (as mentioned in the OP), and tropes like good is stupid, the lawful stupid paladin, and the cleric who charges for healing.
I think this quote is relevant: “Two novels can change a bookish fourteen-year old’s life: The Lord of the Rings and Atlas Shrugged. One is a childish fantasy that often engenders a lifelong obsession with its unbelievable heroes, leading to an emotionally stunted, socially crippled adulthood, unable to deal with the real world. The other involves orcs.”
Stoicism
Though likely the philosophy itself is never name dropped or used, or even known about by most, it seems to be one of the dominant philosophies at gaming tables. This is seen in the adamant refusal to react in anything approaching a realistic way to anything that happens in the game, i.e. the player who refuses to roleplay their character. The player who treats their character as a game piece to be moved from one combat encounter to the next, who speaks of their character in the third person, and will only ever describe what the character does, and avoid ever speaking in character or otherwise engaging in the roleplay.
Taoism
Though it's really a Chinese religion vastly more complicated than most Westerners will ever realize, Taoism is treated as a kind of pop philosophy in the West. I think we see this most often with druids, monks, and other characters playing true neutral. With its flow state and balance of nature, it's a great fit for that let it go, don't grasp, go with the flow attitude. Admittedly, quite a few druids don't play that way, they go full ecowarrior (I've done it myself because it's fun). Taoism influenced Buddhism and so became Chan Buddhism, and when it migrated into Japan was called Zen Buddhism. I see a lot of monks going for a Zen philosophy.
Absurdism and Pessimism are two I'd like to see make an appearance. A Pessimistic cleric. "Hello, fellow-sufferer."
