<p>Is a Star-Schema design essential to a data warehouse?  Or can you do data warehousing with another design pattern?</p>

The thing about star schemas is they are a natural model for the kinds of things most people want to do with a data warehouse. For instance it is easy to produce reports with different levels of granularity (month or day or year for example). It is also efficient to insert typical business data into a star schema, again a common and important feature of a data warehouse.
You certainly can use any kind of database you want but unless you know your business domain very well it is likely that your reports will not run as efficiently as they could if you had used a star schema.