Monday: London on Board hosts Monday night gaming at the Red Herring, Gresham Street from 5.30pm (free to play).
Tuesday: Explore patterns in prime numbers at this Royal Society public lecture from 6.30pm (free).
Friday: Best of Britannia celebrates British made things, music, talent and innovation at the Farmiloe Building, Clerkenwell (tickets £5).
Saturday: Explore Elephant & Castle during Elefest with lots of free art and community events all weekend (free).
Visit artists' open studios in NW10 on Saturday and Sunday between 1-6pm (free).
Sunday: Taste, buy, craft and see the Corne Queen at Apple Day at Roots and Shoots in Kennington from 11am-4pm (£1 adults, kids free).
It's also fruity at the London Orchard Festival in Camley Street Natural Park, Kings Cross from 10.30am-7pm (free).
Check out our things to do in London for free page for more.