It depends on the country. There is no one answer. Except to go read some guidebooks.

Which countries support such a travel style? Where I can rent a car and just... go. Nothing in front of me but the open road.

Australia, right?

More or less all the countries under the Schengen Visa?

No idea on India, Russia, Asia in general (in Japan you can apparently), Africa, and South America.

Seems like the biggest limitation is internet connection. You can arrange coverage before you leave the US. Should be similar to a roaming cell phone data plan(already available in numerous locations), correct?

Well, decent internet service is available in most countries but like in the US there are "pockets" where you'll find limited coverage. I've traveled around the world and rarely have had difficulty finding "internet hot spots". In the last 10 years I can't recall ever staying in a hotel/motel that didn't have internet - however the service might not always have been free. Many restaurants, bars, pubs, coffee shops, etc. have free internet - it's no different than what you found traveling around the US.

You can also sign up for international service on your cell phone and purchase their data plans. Note: Depending on your provider and the providers in the countries you are visiting you may pay hefty Roaming Fees but you will still be able to access the internet almost anywhere you go. For example we recently visited England, Scotland, Ireland, Iceland, the Shetland Islands, the Faroe islands, Norway, Sweden, Denmark, and several russian countries along the Baltic and always were able to get internet and 3G/4G service. 5 years ago we literally traveled around the world (20 countries on 5 continents) and had no difficulty accessing the internet nor using our cell phones.

As far as being able to travel "on your own" you can do that almost everywhere with the possible exception of the Russian countries. They have strict visa and travel regulations making individual travel (without a guide) somewhat difficult so you'll have to do more research if you want to visit those countries.

As a US citizen you can go to the US State Department website (http://travel.state.gov/travel/travel_1744.htmland) to research the basic travel and visa requirements for just about every country. You should then visit the "official" government website for the particular countries you want to visit for more specific info.

Good luck, have fun in your travels.

If you don't make being able to rent a car and drive yourself a requirement and if you don't make camping a requirement, you can travel much of the world. A number of Asian countries prohibit camping, and you can't rent a car and self-drive in some, like China. And you wouldn't want to self-drive even if you could in, say, India. And, of course, there are areas where internet access is difficult, like Burma or Sikkim. Most of the world now has internet access, though in some places it is so slow you may not be able to do your work over those connections.

Except for the car & the camping as the method, the rest of what you describe pretty much fits any round-the-world backpacker.

There's no other country in the world, apart from the country you are a citizen of, where "stay for however long I wanted". Most tourist visa only permit a certain number of days in the country itself even if the life of the visas is several years.

Secondly Schengen visa only cover certain countries in Europe and Australia is not in Europe.

Hanuman,

I think what the OP meant by "stay for however long I wanted" is that she didn't have to worry about meeting an external schedule. E.g. if she was visiting a certain area she really liked and decided to stay an extra week, she could without worrying about transportation (train, plane) schedules or hotel reservations. And of course this could not be done if on a tour.

Also, she wasn't implying that Australia was part of Europe. Not sure how you read that from the post...

"And of course this could not be done if on a tour."

Huh? Hadn't noticed anyone suggesting a tour.

Transportation plans can be changed. In most places they can easily be made at the last minute, although often at extra cost. What can't be changed are visa regulations.

If you have time and plenty of money it would be easy. Except for the time limit requirements imposed by each country you can show up and kick around however you like. In Europe it would more likely be by train and staying in hostels, in Australia rent a camper van, etc. etc. You'd just need to do your research up front to see about local forms of transportations for the places you want to get to, whether that's local busses, trains, or car rental works best.

You'd likely be better off to figure out a country/region that you're interested in and then look for information for that specific area on the forum dedicated to that country/continent.