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.
Tucked away down a cobbled alley behind Russell Square Station, The Horse Hospital, a converted stable, is a unique and totally independent arts space specialising in underground and experimental media. The gallery space itself (known as The Chamber of Pop Culture) houses unusual exhibitions, film screenings and other events.
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.
I grew up in Brighton which is full of cheesy bars like this. If you can handle being surrounded by multiple screens playing Kylie at full volume, its pretty much the cheapest bar in central london. The cliental is so varied its great for people watching and completely un pretentious.