> Another idea would be to convert some C extensions of the standard library to the limited C API. It would ensure that the limited C API contains enough functions to be useful, but would also notify us directly if the API is broken.

First issues that I met when I tried that:

* C code generated by Argument Clinic is incompatible the limited C API: METH_FASTCALL, _PyArg_CheckPositional(), static _PyArg_Parser, etc. are excluded from the limited C API.
* PyTypeObject is opaque and so it's not possible to implement a deallocator function (tp_dealloc) which calls tp_free like:
  Py_TYPE(self)->tp_free((PyObject*)self);
* _Py_IDENTIFIER() is not part of the limited C API