Free Museums/Theatres

All of London's main museums and galleries are free to visit, apart from the occasional temporary exhibition. There are too many to list but it is enough to say you can visit all of the best known without charge - The National Gallery, the Tate, the National Portrait Gallery, the Natural History Museum, the Museum of London and the Victoria and Albert Museum. If your tastes are a bit less mainstream you should be able to find something to suit you and still not pay: the Kew Bridge Steam Museum, the Ragged School Museum, the Clowns Museum and the Dental History Museum are all free too. You could visit one free museum in London every day for weeks.
If you are interested in music you could consider one of the free organ music recitals held at lunchtimes in many of London's churches - Westminster Abbey, St. Giles-in-the-Fields and the church of St. James in Piccadilly are all good places to try and are easy to reach. For more diversity there are lunchtime concerts of all kinds of music at the Guildhall School of Music and Drama at the Barbican.
Lots of pubs have live rock, folk and country bands playing and often don't charge for entry; if you were going to the pub anyway, doesn't this count as free? Some of the London parks stage free music events during the summer and there are longer free festivals too.
Another expensive London pastime is theatre-going and, admittedly, it is more difficult to do this on a budget; however, students may be able to get free entry to drama school productions, or may even be lucky enough to bag one of the limited number of freebies for previews that are put aside for drama students.
If you are prepared to get your hand in your pocket and shell out a whole Ten Pence you may be able to get tickets for performances at the Royal Court Theatre but it will be a standing place with restricted viewing. Other fringe theatres have "Pay what you can nights" so, again, not free but close.