You know one experience that's uniquely London?
Sitting in the front seat of the top deck of a bright red double-decker bus. This is by far the most coveted of all the seats, such that every time I score it, I feel as though I've done something to deserve it!
I imagine it as the equivalent of sitting in an airplane cockpit--albeit without the actual flying part. Still, for only £1.50, you get a pretty nice view. You can look down on people, shops, and the world below. You can look straight into second-story windows and observe people working and living. You can wave at fellow privileged top-deckers in oncoming buses. (They'll most likely ignore you, but hey, spread the love!)
Plus, the buses themselves are pretty iconic, which doesn't hurt.
It's just a little "Yay, London!" moment in my book. And that's always nice.