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To get back to my original objection, it shouldn't be that difficult to differentiate between "__context__ never set" and "__context__ explicitly suppressed".
(e.g. using a property that sets an internal flag when set from Python code or via PyObject_SetAttr, or else a special ExceptionContextSuppressed singleton).
BaseException could then be given a "no_context" class method that did whatever dancing was necessary to suppress the context.
So Steven's examples would become:
def process(iterable):
 try:
 x = next(iterable)
 except StopIteration:
 raise ValueError.no_context("can't process empty iterable")
 continue_processing()
def func(x):
 try:
 x + 0
 except (ValueError, TypeError):
 raise MyException.no_context('x is not a number')
 do_something_with(x)
With appropriate changes to the exception display code, no_context could be as simple as the C equivalent of the following:
@classmethod
def no_context(cls, *args, **kwds):
 exc = cls(*args, **kwds)
 exc.__context__ = ExceptionContextSuppressed
 return exc
Another alternative would be an additional internal flag queried by the exception chaining code itself:
@classmethod
def no_context(cls, *args, **kwds):
 exc = cls(*args, **kwds)
 exc.__context__ = None
 exc._add_context = False
 return exc