[pydrake] Add example of memory leaks

bindings/pydrake/BUILD.bazel

@@ -392,6 +392,16 @@ drake_py_unittest(
     tags = ["lint"],
 )
 
+# TODO(jwnimmer-tri) Once this a real test, switch it to drake_py_unittest.
+drake_py_binary(
+    name = "memory_leak_test",
+    srcs = ["test/memory_leak_test.py"],
+    add_test_rule = True,
+    deps = [
+        ":all_py",
+    ],
+)
+
 drake_py_binary(
     name = "stubgen",
     srcs = ["stubgen.py"],

bindings/pydrake/test/memory_leak_test.py

@@ -0,0 +1,73 @@
+"""Eventually this program might eventually grow up to be an actual regression
+test for memory leaks, but for now it merely serves to demonstrate such leaks.
+
+Currently, it neither asserts the absence of leaks (i.e., a real test) nor the
+presence of leaks (i.e., an expect-fail test) -- instead, it's a demonstration
+that we can instrument and observe by hand, to gain traction on the problem.
+"""
+
+import dataclasses
+import gc
+import sys
+
+from pydrake.systems.analysis import Simulator
+from pydrake.systems.framework import DiagramBuilder
+from pydrake.systems.primitives import ConstantVectorSource
+
+
+@dataclasses.dataclass
+class RepetitionDetail:
+    """Captures some details of an instrumented run: an iteration counter, and
+    the count of allocated memory blocks."""
+    i: int
+    blocks: int | None = None
+
+
+def _dut_simple_source():
+    """A device under test that creates and destroys a leaf system."""
+    source = ConstantVectorSource([1.0])
+
+
+def _dut_trivial_simulator():
+    """A device under test that creates and destroys a simulator that contains
+    only a single, simple subsystem."""
+    builder = DiagramBuilder()
+    builder.AddSystem(ConstantVectorSource([1.0]))
+    diagram = builder.Build()
+    simulator = Simulator(system=diagram)
+    simulator.AdvanceTo(1.0)
+
+
+def _repeat(*, dut: callable, count: int) -> list[RepetitionDetail]:
+    """Returns the details of calling dut() for count times in a row."""
+    # Pre-allocate all of our return values.
+    details = [RepetitionDetail(i=i) for i in range(count)]
+    gc.collect()
+    tare_blocks = sys.getallocatedblocks()
+    # Call the dut repeatedly, keeping stats as we go.
+    for i in range(count):
+        dut()
+        gc.collect()
+        details[i].blocks = sys.getallocatedblocks() - tare_blocks
+    return details
+
+
+def _main():
+    # TODO(jwnimmer-tri) Add command line flags for selecting which duts to
+    # run, change the count, etc. For now, this just hard-codes some things.
+    all_duts = dict([
+        (dut.__name__, dut)
+        for dut in [
+            _dut_simple_source,
+            _dut_trivial_simulator,
+        ]
+    ])
+    for name, dut in all_duts.items():
+        details = _repeat(dut=dut, count=25)
+        print(f"RUNNING: {name[1:]}")
+        for x in details:
+            print(x)
+
+
+assert __name__ == "__main__", __name__
+sys.exit(_main())