Background and Legacies
Outside the dungeons, pits, castles and sewers there is the chance to explore what the characters are more than just damage dealing machines, what they want to be their legacies, who they know, what they want to be; encourage players to expand on their backstories - or examine their backgrounds and see what can come out of the woodwork, not even as a battle but as someone to talk to or stories that they can find out about.
High level characters should have not a small amount of fame and NPCs should be clamouring to apprentice/advise/woo/get money/kill them depending on what they've become famous for. Think Wizard duels to determine "the greatest magi of the land", building sage towers to tutor a school of magic, creating new religious movements, creating a guild of adventurers to solve all the petty tasks that no longer are worth the time, etc.
At Epic level politics becomes an important factor; characters who are powerful can solve important problems, defeat enemies or swing wars between countries and Kings/Nobles/Wizards will want the PCs on their side - or rid of them - for wider political motives
Think scheming political figures "Have this castle as my thank you for your service" (said castle is crawling with dangerous bandits) religious orders that are asking for help with crusades and so on.
Making it alive
And those the players know will have moved on as well; NPCs shouldn't stand still, give a little snippet to important figures in the characters lives to make things feel more "alive", Bob and Bobette have had a baby, Duke Von Bob has created a new trade route, Trader Bob has partnered with his former enemy anti-Bob.
A world of shops and taverns?
Another thing to help the players feel a place is more alive is to add things to the city that are more than rumour-generation taverns and item-trading shops. Don't scattergun these in, you can drip feed them schools, colleges, magical research centres, masons guilds, graveyards and religious institutes to the point that (hopefully) they'll ask "is there an X" in town and then you can sit back and stuff in some generic NPCs for X, give it a name and have them talk to a piece of the city that they've effectively generated for you. Now that's lazy GM'ing and I like that :)
PC: "Is there an alchemist in the city?"
Me: (Checks list of random npc names, rolls a dice for personality on a quick chart), writes 'Alchemist, gruff, slightly mad, level 15' "Yes there is, a few inquiries tells you about Old Bob the Alchemists shop of chemical wonders, it's down old goat street."