1Museums: There are of course the usual: British Museum, V&A and the Science Museum. But try out the Museum of Childhood at Bethnal Green, the Museum of London and the Petrie Museum of Egyptian Archaeology for a bit of a change. They're all free! (But a donation is always welcome.)
2Parks: Hampstead Heath is an enormous park to get lost in, Hyde Park has the Serpentine River, Holland Park includes the beautiful Japanese-inspired Kyoto Gardens. My favourite, however, is Regents Park as there always seems to be something going on there. This sculpture in the free Frieze Art Fair sculpture garden, for instance.
3Art: The National Portrait Gallery, the Tate Britain, Tate Modern, National Gallery are all wondrous art galleries. But I would urge you to see the Whitechapel Art Gallery that you can visit as an add-on to a trip to Brick Lane (check opening times before you go) and the White Cube galleries in Bermondsey and Hoxton.
4Talks: Gresham College offers a range of science and history talks at lunchtimes and in the evening. The Wellcome collection hold evening events on a host of subjects, all vaguely connected with medicine. And if art is your thing, Tate Modern have a host of free films and talks for the discerning freebie hunter.
5Music: Keep an eye on Time Out's music listings for the latest free music events in London. The London Jazz Festival (held annually) is an example of an event that incorporates quite a few free gigs.
6South Bank: This is an area of London where it would be hard not to find something going on. You get free music and art exhibitions and there's often a food market of some sort at the Southbank Centre. I've watched a great French jazz band on the Ground Floor of the National Theatre before heading upstairs to watch a play.
7Street markets: These areas are great to visit for the atmosphere alone - by all means buy something, but it's not strictly necessary. Broadway Market is becoming more and more popular and if you're a young hipster you'll fit right in; Columbia Road Flower Market is worth a visit to listen to the calls of the traders (and if you arrive towards lunch-time you'll find you get more flowers for your money.) The markets on Brick Lane on a Sunday are well worth a visit as well, but watch your mobile phone and MP3 players - I had mine stolen a couple of weeks ago. 8Comedy nights: The Absolutely Free Comedy Night in Soho is held twice a week, and you may get to see one of the bigger names in comedy at it. Angel Comedy in the Camden Head, Islington provides great entertainment on a Saturday night. And I love the title of this one: "Lewis Schaffer is Free Until Famous" is London's longest-running sold UK show. Lewis performs in The Source Below, Soho two or tree times a week.
9TV and Radio recordings: Look no further than TV Recordings for tickets to free recordings in London.
10Free walking tour: Go on a free Royal London tour with Sandemans New Europe. The tour guides don't get paid, so they rely on tips. But this means that you can give according to your budget. Or go on a self-guided tour.
Julia what about elaborating on the free radio recordings? If you go to the BBC tickets website athttp://www.bbc.co.uk/showsandtours/tickets/ you can check to see what's on offer (including TV shows). You can also list your particular preferences so that they will send you a reminder whenever there are tickets available for your favourites. I've been to quite a few recordings this way, although some of the more popular ones are heavily subscribed and there's no guarantee that you'll get tickets.