1. Swim in a Lido.
2. Eat your way around Borough Market.
Go wild on the samples. This post has a handy list of delicious items for under a fiver here.
3. Perform Hip Hop Karaoke at the Social.
4. Go up Big Ben.
You can climb the Elizabeth Tower for free, and go behind the clock faces. Be advised that you're best off booking this six months in advance, and you will have to be a UK citizen. Find out more here.
5. Go Geocaching.
Follow a map to pre-set co-ordinates to find hidden treasure. Download the official Geochaching app and get exploring London's secrets.
6. Watch an opera at The Royal Opera House.
Catch a ballet or the opera for under £5. Tickets are priced from £4 if you don't mind standing in the slips.
Find out more here.
7. Visit Columbia Road Flower Market.
You can grab a cheerful bunch of daffodils for under £5 easily, and if you leave it until later in the day, traders will give you a decent deal on armfuls of blooms.
8. See a show at The Royal Albert Hall.
If you're canny, you can buy tickets for the London Proms from £5. Here's a good how-to.
9. Visit a City Farm.
10. Hire a deck chair in the park.
Most city parks will offer deckchair hire in the summer. For example, in Green Park, you can rent a chair up to three hours for under a fiver. Find out more here.
11. Be a Groundling at The Globe.
If you don't mind standing, you can see some fine acting for pocket change. Find out more here.
12. Visit the Houses of Parliament.
14. Visit The Morpeth Arms Spying Room.
For the price of a pint, the upstairs lounge at the Morpeth Arms allows you to peer over the River Thames, with free telescopes and binoculars. If you're keen, there are also views over the M16 Building. Find out more here.
15. Stay late at The Science Museum.
16. Do the Fulham Palace parkrun.
If you're a keen runner, there is a free weekly timed 5K run around Fulham Palace Park every Saturday. Not half bad, eh? Find out more here.
17. Climb The Monument.
Climb the Monument for £3 - and get a certificate for your troubles. Find out more here.
18. Cruise on the London Waterbus.
Enjoy a 35 minute cruise for a fiver - if you plan your route right. Trips from London Zoo to Little Venice cost £5. Find out more here.
19. Watch a film At The Coronet Cinema.
H/t to this Reddit thread.