How do you do TDD in a non-trivial application?

It's easier than you think, actually. You just use TDD on each individual class. Every public method that you have in the class should be tested for all possible outcomes. So the "proof of concept" TDD examples you see can also be used in a relatively large application which has many hundreds of classes.

Another TDD strategy you could use is simulating application test runs themselves, by encapsulating the main app behavior. For example, I have written a framework (in C++, but this should apply to any OO language) which represents an application. There are abstract classes for initialization, the main runloop, and shutting down. So my main() method looks something like this:

int main(int argc, char *argv[]) {
  int result = 0;

  myApp &mw = getApp(); // Singleton method to return main app instance
  if(mw.initialize(argc, argv) == kErrorNone) {
    result = mw.run();
  }

  mw.shutdown();
  return(result);
}

The advantage of doing this is twofold. First of all, all of the main application functionality can be compiled into a static library, which is then linked against both the test suite and this main.cpp stub file. Second, it means that I can simulate entire "runs" of the main application by creating arrays for argc & argv[], and then simulating what would happen in main(). We use this process to test lots of real-world functionality to make sure that the application generates exactly what it's supposed to do given a certain real-world corpus of input data and command-line arguments.

Now, you're probably wondering how this would change for an application which has a real GUI, web-based interface, or whatever. To that, I would simply say to use mock-ups to test these aspects of the program.

But in short, my advice boils down to this: break down your test cases to the smallest level, then start looking upwards. Eventually the test suite will throw them all together, and you'll end up wi