Not the most exciting place inside, but a lovely rooftop area which not all that many people seem to know about, 2 mins walk from Old St station. If it’s packed then The Red Lion is just up the road.
The Greenwich Observatory with its green laser beam can transfigure any night sky. If you decide to cross the Thames via the foot tunnel, make sure you look out for the green ray. You can be on one side of the meridian in a minute, and cross to the other side on the next, without even realising it.
One of my favourite things about London is the underground transport system itself. I love the design of the information systems and the stations have so much history — some of them dating right back to the 1800s.
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.