Barbican Centre
The Barbican Centre is the The Corporation of London's main cultural venue. It sits in the middle of a maze of residential concrete - the whole estate (known collectively just as 'The Barbican') seemingly designed as one of those Brave New Worlds of culture and harmonious living. The tower blocks themselves soar about 40 stories above Aldersgate.
It has two cinemas, playing both arthouse and major releases, two theatres, a large hall, and a gallery. It's also home of the LSO.
If you go even remotely regularly, it's worth getting an annual Barbican card for 10, for which you get the guide sent every month, 1.50 off the cinema and 20% or so off a lot of the tickets.
The Barbican Centre also contains the Museum of London, and looks out on a small bit of Roman wall. There are also two pubs and a Pizza Express in the complex.
Navigation is a little nightmarish, both to find the place and once inside. The steps and mazes and poor signposting within mirror those without.