The overflowed flag was introduced 8 years ago (near the release of Python 3.0) by the following changeset:

changeset:   42013:cd125fe83051
user:        Martin v. Löwis <[email protected]>
date:        Sun Jun 10 09:51:05 2007 +0000
files:       Include/ceval.h Include/pystate.h Include/stringobject.h Include/unicodeobject.h Lib/test/test_frozen.py Lib/test/test_new.py Lib/test/test
description:
Make identifiers str (not str8) objects throughout.
This affects the parser, various object implementations,
and all places that put identifiers into C string literals.

In testing, a number of crashes occurred as code would
fail when the recursion limit was reached (such as the
Unicode interning dictionary having key/value pairs where
key is not value). To solve these, I added an overflowed
flag, which allows for 50 more recursions after the
limit was reached and the exception was raised, and
a recursion_critical flag, which indicates that recursion
absolutely must be allowed, i.e. that a certain call
must not cause a stack overflow exception.

There are still some places where both str and str8 are
accepted as identifiers; these should eventually be
removed.