Yes, they do exist. I have a whole team of full-stack developers. I gave one an iphone application to fix bugs in and convert from ios5 to ios7. He had no prior objective-c experience and in a week and a half, he had fixed all of the bugs and even dove into a 3rd party framework we were using to describe a problem to me using it with ios7.
A full-stack to me is somebody that can do web development, do server-side development, optimize SQL queries, understand security, scalability, and able to build linux server infrastructures for large software systems. They also are able to take any technology and provide value to a business using it in a matter of a couple of weeks.
That's the best definition of for "full-stack" developer I've seen. There are two big things that set these people apart from their peers:
* Willingness to dive in and figure things out.
* Having a wide enough knowledge base that they know where to and what to look for when they don't know something.
How? Especially if they've never learned the language/framework? This "Paul Bunyan" fairy tale of the "just do it!" hacker who can solve any problem no matter the odds just doesn't exist in real life.
Hello. I am this person. I am a generalist, I am not the best in any one area, but I have a broad range of skills across many domains. If I don't know how to solve a new problem, perhaps in a new domain, I will leverage the broad knowledge and learning skills to find out how to solve the problem. I won't be as fast as a specialist and there will be a greater risk of failure, but there is a good chance I will get the job done. I am not arrogant, I don't think I'm a genius, I don't think I am infallible. It's just that I am a generalist who likes to know a little about a lot, it's the way my brain works.
Learning just enough of a language or a framework to patch up a solution to a specific problem doesn't take very long. It takes a bit of experience, but it doesn't take a mythical hyper-productive hacker. The solution might not be pretty, and it probably won't scale to millions of users or stay solid for decades, but it will solve the problem. And in many cases, it's all that matters.
Even if you've never seen the language/framework, you dive in and learn it in an efficient and effective way, both from the current codebase and other resources.
Maybe you just haven't met the right people. I definitely "just do it" all the time. When I needed to develop a single page app for the first time, I just did it. When I found I needed to automate server deployments, I just did it. When a project I told someone I'd prototype needed me to design a circuit, fabricate a PCB, build it by hand and write firmware for its ARM microcontroller (including an OS), I just did it. And had fun doing it.
I'm sticking with my "full stack" claim. I think I've earned it.
A full-stack developer is a developer who sees something that needs to get done and does it, even if they currently don't "know how".
And these people do exist, and are every bit as valuable as you'd expect.