We used to have this game in the old DirtyWarez, it may be considered to belong to the more intellectual variety.
You answer the previous question and then ask your own question.
Example:
(Previous question
Why did the chicken cross the road?
Because he was approaching a KFC and didn't want to take any chances.
(And now my question
Why do cats like mice?
You answer the previous question and then ask your own question.
Example:
(Previous question
Because he was approaching a KFC and didn't want to take any chances.
(And now my question