How about Scrabble? I know it's not exactly a 'reading' game as such, but you have to be able to read to make a good game. Or there's Monopoly, a grand old US game eveybody plays in different forms all over, that involves reading and arithmetic (how many hotels can you afford to build on Park Lane/South Michigan Avenue before you turn into a tycoon and test everybody's friendship?)
Or even better: Consequences. Everybody sets off with a clean sheet of notepaper, writes something at the top, turns over the paper and passes it on to their right, the next adds another phrase or line and so on in a circle. You tell everybody basically what to write (what they said/did/wanted to say etc) and at each turn of the paper something else (what colour sheets they had on their bed) and so on.
At the end:
'Sam said shall we go for a coffee,
We went to Buckingam Palace
we had pizza on the front lawn (?)
and went up to bed.
Jack had red and blue candy striped sheets
and I dreamed of having a drink
with Henry VIII'
Take it where you want...