I am going to suggest that there are only two kitchen appliances that are absolutely necessary in a person's first kitchen: a fridge and a stove. Do note that this series isn't about "the things that you need to get by in your dorm room in college"; this is a series about YOUR FIRST KITCHEN in your first house or apartment. The series presumes responsibility for your own food and your own diet and your own cooking.
Even if a person is intending to shop daily for food, a kitchen still needs cold food storage. Any reasonably healthy diet in this part of the world needs dairy and fresh produce, all of which need cold storage. Fortunately, rental spaces usually come with one and houses are built to accommodate one. Whether a fridge NEEDS a freezer or not is dependent, I think, on personal taste and on a person's shopping style. If you like ice cream, you're gonna need a freezer. If you want to buy things like meat in bulk to last for weeks and weeks, you're gonna need a freezer. If, on the other hand, you hate ice cream and you are going to shop every day for fresh food, then maybe you don't need a freezer. But the fridge, the fridge you need.
A person also needs a stove. It's tough to cook without heat and the stove (incl. oven) is the most versatile appliance out there for heating food. You can cook everything on a stove or in an oven, and in fact, that's where you SHOULD cook them. You can't saute an onion in a microwave and even if you could, why would you want to?
That's not to say that a good microwave or a toaster oven isn't useful and I think that most people would consider a microwave to be a pretty standard addition to a kitchen. I think I'll save further discussion about small(er) appliances for another day as this post has gone on long enough already.
Thoughts? Comments? Disagreements? Ringing endorsements? Please contribute in the comments!