You can not $expand
any lookup column of child list. You can only $expand
all lookup column of parent list. Let say, you are querying on A list and A has a lookup column from B list. B has a lookup column from C list. So if you are querying on A list, then you can only $expand
all columns of B.
/_api/web/lists/getbytitle('A')
You can not $expand
any column from C. The reason is there is no direct relation between A and C.
Workaround
At first get the item Id
of B list, then make another REST call to B and $expand the columns of C
/_api/web/lists/getbytitle('B')/items({ItemId})?$select={select column from c}&$expand=C
You can use SharePoint REST Client to test your query.
Update
You can make a batch request as you have many items from B.
Make batch requests with the REST APIs