diff --git a/Lib/test/test_typing.py b/Lib/test/test_typing.py
index 58566c4bfc8..c3a092f3af3 100644
--- a/Lib/test/test_typing.py
+++ b/Lib/test/test_typing.py
@@ -4323,6 +4323,16 @@ class GenericTests(BaseTestCase):
         c.bar = 'abc'
         self.assertEqual(c.__dict__, {'bar': 'abc'})
 
+    def test_setattr_exceptions(self):
+        class Immutable[T]:
+            def __setattr__(self, key, value):
+                raise RuntimeError("immutable")
+
+        # gh-115165: This used to cause RuntimeError to be raised
+        # when we tried to set `__orig_class__` on the `Immutable` instance
+        # returned by the `Immutable[int]()` call
+        self.assertIsInstance(Immutable[int](), Immutable)
+
     def test_subscripted_generics_as_proxies(self):
         T = TypeVar('T')
         class C(Generic[T]):
@@ -8561,6 +8571,17 @@ class AnnotatedTests(BaseTestCase):
         self.assertEqual(MyCount([4, 4, 5]), {4: 2, 5: 1})
         self.assertEqual(MyCount[int]([4, 4, 5]), {4: 2, 5: 1})
 
+    def test_instantiate_immutable(self):
+        class C:
+            def __setattr__(self, key, value):
+                raise Exception("should be ignored")
+
+        A = Annotated[C, "a decoration"]
+        # gh-115165: This used to cause RuntimeError to be raised
+        # when we tried to set `__orig_class__` on the `C` instance
+        # returned by the `A()` call
+        self.assertIsInstance(A(), C)
+
     def test_cannot_instantiate_forward(self):
         A = Annotated["int", (5, 6)]
         with self.assertRaises(TypeError):
diff --git a/Lib/typing.py b/Lib/typing.py
index 347373f0095..914ddeaf504 100644
--- a/Lib/typing.py
+++ b/Lib/typing.py
@@ -1127,7 +1127,9 @@ class _BaseGenericAlias(_Final, _root=True):
         result = self.__origin__(*args, **kwargs)
         try:
             result.__orig_class__ = self
-        except AttributeError:
+        # Some objects raise TypeError (or something even more exotic)
+        # if you try to set attributes on them; we guard against that here
+        except Exception:
             pass
         return result
 
diff --git a/Misc/NEWS.d/next/Library/2024-02-09-07-20-16.gh-issue-115165.yfJLXA.rst b/Misc/NEWS.d/next/Library/2024-02-09-07-20-16.gh-issue-115165.yfJLXA.rst
new file mode 100644
index 00000000000..73d3d001f07
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2024-02-09-07-20-16.gh-issue-115165.yfJLXA.rst
@@ -0,0 +1,4 @@
+Most exceptions are now ignored when attempting to set the ``__orig_class__``
+attribute on objects returned when calling :mod:`typing` generic aliases
+(including generic aliases created using :data:`typing.Annotated`).
+Previously only :exc:`AttributeError`` was ignored. Patch by Dave Shawley.