I like Greenwich, but it's a tourist trap sadly. I only go in winter when there aren't that many cluttering up the streets. Love the market, but only for mooching, far too inflated prices for bric-a-brac
But I really love Southwark, or suvvark as we'd say! Once you get off the beaten track, down the borough, or even down Tooley Street, mind you the Mayors testicle Parlour is there so it's a bit more on the track these days, I remember when all the rail way arches were pubs
and Hays Galleria was in its infancy.
So many good pubs in, brilliant one, right down the Borough, called the Mermaid, ooh the fish lunches we used to have.
Things like Vinopolis ruin it. Fantastic pizza parlour in Southwark Cathederal, and if it's your thing, there is always a lunchtime musical.
I also don't think you can beat St Martin in the Fields up on T/Sq either.
I think I could do you every pub in The City too 
is well old