"PyPy is a reimplementation of Python in Python" is a rather misleading way to describe PyPy, IMHO, although it's technically true.
There are two major parts of PyPy.
- The translation framework
- The interpreter
The translation framework is a compiler. It compiles RPython code down to C (or other targets), automatically adding in aspects such as garbage collection and a JIT compiler. It cannot handle arbitrary Python code, only RPython.
RPython is a subset of normal Python; all RPython code is Python code, but not the other way around. There is no formal definition of RPython, because RPython is basically just "the subset of Python that can be translated by PyPy's translation framework". But in order to be translated, RPython code has to be statically typed (the types are inferred, you don't declare them, but it's still strictly one type per variable), and you can't do things like declaring/modifying functions/classes at runtime either.
The interpreter then is a normal Python interpreter written in RPython.
Because RPython code is normal Python code, you can run it on any Python interpreter. But none of PyPy's speed claims come from running it that way; this is just for a rapid test cycle, because translating the interpreter takes a long time.
With that understood, it should be immediately obvious that speculations about PyPyPy or PyPyPyPy don't actually make any sense. You have an interpreter written in RPython. You translate it to C code that executes Python quickly. There the process stops; there's no more RPython to speed up by processing it again.
So "How is it possible for PyPy to be faster than CPython" also becomes fairly obvious. PyPy has a better implementation, including a JIT compiler (it's generally not quite as fast without the JIT compiler, I believe, which means PyPy is only faster for programs susceptible to JIT-compilation). CPython was never designed to be a highly optimising implementation of the Python language (though they do try to make it a highly optimised implementation, if you follow the difference).
The really innovative bit of the PyPy project is that they don't write sophisticated GC schemes or JIT compilers by hand. They write the interpreter relatively straightforwardly in RPython, and for all RPython is lower level than Python it's still an object-oriented garbage collected language, much more high level than C. Then the translation framework automatically adds things like GC and JIT. So the translation framework is a huge effort, but it applies equally well to the PyPy python interpreter however they change their implementation, allowing for much more freedom in experimentation to improve performance (without worrying about introducing GC bugs or updating the JIT compiler to cope with the changes). It also means when they get around to implementing a Python3 interpreter, it will automatically get the same benefits. And any other interpreters written with the PyPy framework (of which there are a number at varying stages of polish). And all interpreters using the PyPy framework automatically support all platforms supported by the framework.
So the true benefit of the PyPy project is to separate out (as much as possible) all the parts of implementing an efficient platform-independent interpreter for a dynamic language. And then come up with one good implementation of them in one place, that can be re-used across many interpreters. That's not an immediate win like "my Python program runs faster now", but it's a great prospect for the future.
And it can run your Python program faster (maybe).
