US stockbrokers that accept international clients

I was recently looking for US stockbrokers who are able to open accounts for foreigners (non-resident aliens) from most countries. Surprisingly few brokers are willing to open an account for non-US residents, and some of those who do accept international clients only accept those who live in a small list of countries. Perhaps the paperwork and administrative burden for accepting international clients is too high for most brokers. Here I will be focussing on those US stockbrokers who offer their services to a nearly global client base. In other words, those who can open accounts for residents of most countries of the world.