In <PERSON> Culture novels, society is run by hyper-intelligent AIs called Minds (with a capital M). These seem to do the following things to occupy their time
* practical service, eg running a starship or artificial habitat.
* research, eg let's think as hard as we can about how to make things better for our society. This would include research into weapons or other technologies.
* plot against other factions of Minds, or against alien civilisations/organisations (which they may often believe is also a form of service to their society). 322
As with human politics/intelligence services, these plots vary hugely in scope, timescale and legality.
* mathematical "what if" games, done as a form of recreation: for example, imagining a universe run from different mathematics to our universe, and travelling within this imaginary universe, exploring the consequences of the decisions they made when setting it up. This is referred to as the "Land of Infinite Fun", or the "Land of IF" for short. Some AIs might spend almost all of their time doing this, having effectively withdrawn from society.
* temporary suspension, until needed: this is common with "soldier" minds, who define themselves through combat, and when there is no combat to be done will put themselves into "shutdown" until the next war or nearby skirmish.
EDIT: I should add that individual Minds might divide their time between all of the above (apart from suspension), so that a ship Mind might be carrying on with some arcane plot with other Minds while "on duty", and might also contribute to a research group and dip into the Land of IF in a dull moment. They're extremely good at multitasking, and operate at a speed many orders of magnitude faster than, for example, conversation with a human would require.
EDIT 2: I need to point out that in these novels, the Minds do not see themselves as servants to the humans: the society is nominally an anarchic utopia, with equal citizenship given to Minds, humans and "drones" (sentient robots that are not anywhere near as powerful as a <PERSON>, but which are generally at least as intelligent as a human) but the <PERSON> run the show, and their attitude towards the humans varies between "caretaker", "benevolent dictator", "pet owner", and "(sometimes not so) secretly despising". 44