A walled secret garden, the Physic Garden is like a time capsule, founded in 1673 to allow apprentices of the Worshipful Society of Apothecaries of London to study the healing properties of plants, it is now a living museum. Tours are offered by volunteers with immense knowledge of botany and the cultural history of the garden or visitors can wander the grounds alone and spot odd and fascinating plants such as the mystical Mandrake.
The Hayward Gallery in Southbank center has held my favourite exhibitions in London and each time they are extremely well curated. Definitely a place to visit !
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.
This independent cinema near Angel in North London is over 100 years old and has had several lives including hosting early Sex Pistol gigs. It now serves as my favourite place to watch films (along with the Rio in Dalson).