What you need to take on an assignment depends on your type of housing. Obviously, if you will be housed in an extended stay or corporate housing where all linens and kitchen ware is supplied, this list will be vastly shorter. For the average traveler who is going to be living in the average one-bedroom apartment, furnished with basic furnishings only, you will need to take quite a few things.
Optionally, you can choose to purchase what you need after arrival to the assignment. Dollar stores and garage sales are an easy, cheap and often fun way of providing many of the items that you need. Donating your purchased items after your assignment at the local Goodwill helps the community and provides a tax deduction for you.