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

Eseguire il reproducer fornito per dizionari annidati con Python 3.11 e 3.12, quindi tracciare il modo in cui pickle.dumps gestisce la ricorsione dopo l’aumento di sys.setrecursionlimit. La correzione è completa quando l’esempio viene eseguito correttamente nelle versioni interessate senza RecursionError, preservando il comportamento documentato del limite di ricorsione.