Random Encounters is a writing game that I very recently thought up (and am excited to play soon) and would like to teach to other people. This game basically only requires you to have the equipment needed to write and for you to be familiar with the writing of the other person who is playing with you. By being 'familiar' I mean you just really need to know information about the other writer's settings/characters, something which, I hope, won't be too uncommon. The rest is easy, basically, each person just thinks of one character created by the other person, then when you've each thought of one you tell each other and take it in turns to write a very short story (1,000 words at the most to keep it fun) in which the two characters you've each thought of meet each other. This game will be especially fun if the two of you write in drastically different genres/styles, it'll mean that all kind of fun and outlandish stories will have to be thought up in order for the characters to meet. Plus, it will give you some fun pieces of writing to read back again in the future.
There is a, less fun, version of the game you can play if you know another writer but are not wholly familiar with their work, or if each of you are fanfiction writers. Basically you do it exactly the same, except that you think of any fictional character that your partner will know about and see what you get!
I hope you like my idea!