Adding test case for 80880

Test case is based on reproducer given in ansible#80880, with some minor
adjustments and assertions.

test/integration/targets/handlers/80880.yml

@@ -0,0 +1,34 @@
+---
+- name: Test notification of handlers from other handlers
+  hosts: localhost
+  gather_facts: no
+  handlers:
+    - name: Handler 1
+      debug:
+        msg: Handler 1
+      changed_when: true
+      notify: Handler 2
+      register: handler1_res
+    - name: Handler 2
+      debug:
+        msg: Handler 2
+      changed_when: true
+      notify: Handler 3
+      register: handler2_res
+    - name: Handler 3
+      debug:
+        msg: Handler 3
+      register: handler3_res
+  tasks:
+    - name: Trigger handlers
+      ansible.builtin.debug:
+        msg: Task 1
+      changed_when: true
+      notify: Handler 1
+  post_tasks:
+    - name: Assert results
+      ansible.builtin.assert:
+        that:
+        - "handler1_res is defined and handler1_res is success"
+        - "handler2_res is defined and handler2_res is success"
+        - "handler3_res is defined and handler3_res is success"

test/integration/targets/handlers/runme.sh

@@ -50,6 +50,9 @@ for strategy in linear free; do
   [ "$(ansible-playbook test_force_handlers.yml -i inventory.handlers -v "$@" --tags force_false_in_play --force-handlers \
   | grep -E -o CALLED_HANDLER_. | sort | uniq | xargs)" = "CALLED_HANDLER_B" ]
 
+  # https://github.com/ansible/ansible/pull/80898
+  [ "$(ansible-playbook 80880.yml -i inventory.handlers -vv "$@" 2>&1)" ]
+
   unset ANSIBLE_STRATEGY
 
 done