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.
A place that really grows on you, with each visit the Barbican reveals more of itself to you and your bond is tightened. The Barbican Kitchen is great for coffee or lunch, summer hangouts by the ponds, cinema in winter, the amazing Barbican Hall for live music, and don't miss the conservatory.