Like the fifty states game, trying to think of all thirty-two London boroughs can be harder than it sounds. In fact, I'm far better at naming the fifty US states than I am at naming the boroughs of Greater London. Just like the fifty states game, even three people sat scratching their heads will probably give up before coming up with all thirty-two.
This page is a simple list of the thirty-two London boroughs, easy to view on a mobile phone browser when you're sat at the pub trying to work out which ones you've forgotten.
Note: Inner and Outer London are listed here as defined by the London Government Act 1963, but there are other definitions in use. And the full name of most boroughs begins with "The London Borough of [ . . . ]", with exceptions being the City of Westminster, and the three "Royal" boroughs.
The City of London is not defined as a borough, nor is it within a London borough. It is instead governed by the City of London Corporation.
Once you think you've burnt the full list of thirty-two boroughs into memory, see how good your powers of recall are by playing my London Boroughs Game. I've been developing and testing the game for weeks, and I still tend to get stuck after recalling about twenty-four of the boroughs. And so far I have not once been able to recall the full list before the time runs out. See if you can do better.