Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
There's something so quintessentially London about this hill. The network of paths and lines of lampposts leading you up to a view of the city that's at your fingertips.
Epping Forest is a 2,400 hectares area of ancient woodland between Epping in the north and Wanstead in the south, straddling the border between Greater London and Essex. I tend to get off at Loughton Tube Station (Central Line), which is about 10–12min walk to the forest.