I was wondering about the thread title too.... 
If we're talking about an appliance I can do
without, it'll probably be the dishwasher. Even with the SO over
for meals, I usually don't generate enough dirty dishes, utensils
and cooking implements to justify using the dishwasher. I can wash
and dry them faster in the kitchen sink.
It's hard to say which appliance I can't do
without, as so many of them are essential for daily living. Think
about how downright inconvenient it would be if one didn't have a
stove or refrigerator in the house, let alone a washer or dryer. By
golly, we might as well be back in the Stone Age!