The Wellcome Collection is a free museum and library, mostly dealing with medecine and social history. The exhibitions are unusual and engaging, and the permanent collection includes bizarre items such as medieval masks and human remains.
The Barbican is worth visiting for it's brutalist architecture alone. The concert programming is really diverse; it's home to the BBC Syphony Orchestra, but you'll also find bands like the Dirty Projectors and the National amongst the classical and contemporary composer line up. Plenty of art, dance and theatre too, however, the Conservatory is the real gem that's not to be missed.
Inspired by Dave Eggers’ projects in the USA such as the Pirate Store in San Francisco and Superhero store in Brooklyn, The Monster Supplies Store in Hoxton is home to the Ministry of Stories: A volunteer run organization, founded by Nick Hornby, to help young people with creative writing of all kinds through workshops and free mentoring sessions. The store is the “Purveyor of quality goods for monsters of every kind” with fascinating bottles and tins containing everything from eyeballs to Zadie Smith’s short story Mortal Terror.
I often stop when passing this artery of city traffic. The dwarfing structure casts shapes and shadows which can easily be missed when speeding in/out of the city.