sys.setrecursionlimit no longer affecting pickle.dump

Bug report

Bug description:

This is a follow up of https://github.com/python/cpython/issues/112282 (updating doc).
It's also a different case of https://github.com/python/cpython/issues/128544 (import chains impacted).

In particular, on top of the doc no longer being correct for sys.setrecursionlimit, the regression needs to be fixed.

Note: this works in python 3.11 or below.

import pickle
import sys

nested_object = current = {}
for i in range(10000):
    current['nested'] = {}
    current = current['nested']

sys.setrecursionlimit(3750000)
x = pickle.dumps(nested_object)

Result of the run:

▶▶ python3.11 test-issue.py
# runs ok

▶▶ python3.12 test-issue.py
Traceback (most recent call last):
  File "~/test-issue.py", line 10, in <module>
    x = pickle.dumps(nested_object)
        ^^^^^^^^^^^^^^^^^^^^^^^^^^^
RecursionError: maximum recursion depth exceeded while pickling an object

Impact: all programs trying to pickle data that is too nested cannot be ported to python 3.12 or 3.13 so far

CPython versions tested on:

3.12, 3.13

Operating systems tested on:

Linux, macOS, Windows

Exécutez le reproducer fourni de dictionnaires imbriqués avec Python 3.11 et 3.12, puis suivez la manière dont pickle.dumps gère la récursion après l’augmentation de sys.setrecursionlimit. La correction est complète lorsque l’exemple s’exécute correctement sur les versions concernées sans RecursionError, tout en préservant le comportement documenté de la limite de récursion.