The randomness of the factions' personalities are well-documented with lots of egregious examples. In one example, the faction roles were reversed entirely: Early on, the Spartans declared war as the player was trapping them from getting more land. Fair enough. Then the Peacekeepers made a surprise strike on the player faction even though they were much weaker and the player hadn't declared any wars himself. The Spartans had about as much power as the player, but on the first sign of trouble, they surrendered unconditionally to him. The Peacekeepers, on the hand, continued to fight to the bloody end, refusing to compromise with someone who, again, never started a war. Whoever decided to join that colony for peacekeeping ideals got ripped off to say the least.
The Longevity Vaccine movie features what appears to be a TV show that is a cross between the game and South Park, with a Morgan caricature chasing a Lal caricature away.
The Sound Off example on the main page can be heard when you build a Network Node at one of your bases.
"I don't know but I been told! (I don't know but I been told!) Deirdre's got a Network Node! (Deirdre's got a Network Node!) Likes to press on the on-off switch! (Likes to press on the on-off switch!) Dig that crazy Gaianwitch! (Dig that crazy Gaian witch!)
Sometimes, faction leaders (especially the ones who are neither particularly peaceful nor particularly aggressive) have trouble making up their minds about whether they want a Pact of Brotherhood or not. It can be amusing when Zakharov declares he's had enough of your ecological ramblings/paranoid delusions/pacifistic whining/whatever else your faction leans towards and breaks his pact with you, but is perfectly happy to remake it with no strings attached the very next turn.
It's way too easy to imagine Zakharov's AI as being passive-aggressive.
There was also this one time where the player was trying to make friends with the Data Angels. Roze asked for a loan in return. Fair enough. Then she asked for the player to declare war on another friendly faction. The player said no, fully expecting to be rejected. But then she signed the treaty! It makes sense, though, that Roze would be The Gadfly.
One player had a game where their faction and The Spartans flickered in and out of a Pact every turn for 6 turns. Make up your mind!
Among the options when you start a new game is "Randomize Faction Leader Social Agendas", which replaces each faction's preferred form of social engineering with a random one. Sometimes the results are surprisingly reasonable, such as the Morganites valuing Wealth, but you also get odd cases like the militaristic Spartans wanting you to run a Democracy or the normally anti-scientific Believers demanding that you pursue Knowledge.
Turning down a request from a human (usually for a loan or tech) when you're playing as one of the Progenitor factions will occasionally be phrased as "Human idiom: liar liar pants on fire." There's something distinctly amusing about the powerful, aloof aliens having the tact of an elementary school child.