An arched brick roof is cheap, strong and natural. Many very old buildings with this roof survive to this day.
A living roof is exactly what it sounds like. You have soil as your roof and plants grow to cover it. It is beautiful, keeps your home cool, cheap and natural.
There is almost nothing faster than a metal roof, so if speed is your priority, metal is a great option. However, it is expensive, not particularly durable, heats up in the sun and is loud in the rain.
Latex concrete lends itself to unique and sweeping shapes that you couldn't achieve with many materials. It is cheaper, but slower than metal, and quicker, but more expensive than poured concrete.
A poured concrete roof is about as strong as you can get. You will be able to walk on it without ever thinking twice about falling through.
The roof is where you will lose most of your home's heat, as well as being the place that will heat up the most in summer. So if you want to conserve any energy put in to heating or cooling your house, you will have to pay special attention to insulation.
Metal is expensive, but it is strong and fast. It can be an invaluable item while building your home, having so many different uses.
Catching rain water from your roof is the most accessible source of water off-grid. Everyone gets at least a little rain, and if you size your roof and storage to match your annual rainfall, you need never go without water.