Dynamically add depth / level key to deeply nested JSON object

How to add a key dynamically to each level of a deeply nested object with its level of depth.

The level key should be added dynamically and is not part of the root data.

Expected result should look like this:

{
  level: 0,
  name: 'A',
  id: 9991,
  children: [
    {
      level: 1,
      id: 9992,
      name: 'A A',
      children: [
        {
          level: 2,
          id: 9993,
          name: 'A A A',
        },
      ],
    },
    {
      level: 1,
      id: 9995,
      name: 'A B',
      children: [
        {
          level: 2,
          id: 9997,
          name: 'A B B',
          children: [
            {
              level: 3,
              id: 9998,
              name: 'A B B A',
            },
          ],
        },
      ],
    },
  ],
};