TL;DR Next time you’re in London, book a free ticket to Ceremony of the Keys.
I did it
It’s honestly one of the best touristy things you can do in London!
You’ll get to walk inside the Tower of London (normally £24 for a ticket) when all visitors have left, get to witness one of the oldest ceremonies with beefeaters, talk to the Chief Yeoman Warder — and it’s absolutely free!
What is Ceremony of the Keys
Ceremony of the Keys is held every night at the Tower of London, and has done so in some form or another since the 14th century.
At exactly 9.53pm, the Chief Yeoman Warder, dressed in Tudor Watchcoat, meets the military escort, made up of members of the Tower of London Guard. Together, the Chief Yeoman Warder and the Yeoman Warder ‘Watchman’ secure the main gates of the Tower.
Unfortunately, taking pictures is strictly prohibited during this event. But you can visit multiple times and bring company (just add the number of people into the booking).