I found the poll a bit difficult to answer. Here's why. In my campaign, we do adventuring; virtually everything is revolving around them fighting, travelling to fights, or discussing fights with people. It's fairly combat heavy. However, I also wanted to integrate these demi-god level individuals into the setting, rather than having the oddity of them being so powerful but also having no place in society. So the cleric was given a specific place within the religion as a figurehead pope. He later started a religious order with the party's Monk and Paladin. The Sorceress is an Earl, and was from the start of the game, so regularly spoke to her King in that role, not as an adventurer. The party's Arcane Trickster got knighted and started his own international society of mages to have people to chat to. I facilitated all of this by instituting BIG downtime breaks in the campaign after levels 7 and 14 roughly, where five and ten years passed respectively. That meant that the high level and ultra-dangerous Dwarven Monk is now known widely as the feared but respected Master of the Order of the Holy Fist, and when he goes to speak to the Dwarven High King the court can respond to him in that light.
None of this has a lot of rules attached: we use the DMG guidelines for stronghold building and expenses to sort of eyeball how much it costs to run a 100 monk monastery for a decade, for example, but I don't go about using random event tables or the like. We have had a session each time for sorting out how downtime goes, but otherwise things move as a normal game.
As an explanation for why I was so keen to do this, I refer you to Skyrim. In that, I did the whole main quest, won the civil war, killed a dozen dragons, whatever. And then I decided to do the companions quest, and when I walked in the room one of them said, "I don't know who you are" really dismissively. It drove me mad; there is no way that sentence makes sense in the story that the game has told thus far. I didn't want to have this happen in my campaign, where the King can say, "Wait, who are you guys again?" to the party who could kill everyone in the room in six seconds if they wanted.