London's free museums are a global anomaly — the British Museum, Natural History Museum, Victoria & Albert, Science Museum, Tate Modern, National Gallery, and National Portrait Gallery are all free. This alone justifies multiple days. The British Museum's Egyptian collection, Elgin Marbles, and Rosetta Stone are among the world's greatest artifacts.
Tower of London (Crown Jewels), Buckingham Palace (State Rooms in summer), St. Paul's Cathedral dome climb, Westminster Abbey, and the Houses of Parliament round out historic London. The South Bank walk (from Tate Modern to Borough Market) is a perfect afternoon. Hampton Court Palace, Kew Gardens, and Greenwich with the Prime Meridian are excellent day trips.