3 questions;

What are some good written exercises for advanced students?

What are sme good written exercises for intermediate students?

What are some good written exercises for beginner students?

You can use caricatures. They are fun and encourging. You can talk about them with your students first and want them to summarize it or you can integrate it with the other skills such as reading or listening. For example; one of the students tries to complete the story or make up a new one, the others listen to him/her at the same time, and then anothe one/group continues.
Here's my progression, but it's only very general:

Beginners exchange information.

Intermediates tell stories.

Anvanced students analyze and give opinions.

My typical writing assignments follow these rules. I also usually ask students to write from what they know, not from what they research.
 Doll's reply was promoted to an answer.