It is possible to created a field named '$' with the update command.

let:
db.menu.update(

{'children.children.label': 'New sub menu'}

, {$push: {'children.$.$.children':

{label: 'New new sub menu', children: []}

}})

where all 'childen' fields are arrays.

Expected result:
the item is pushed at in the right sub children matched by the criteria and 2 level deep because of the positional operator.

Actual Result:
{ "children" : [
{
"$" : {
"children" : [

{ "label" : "New new sub menu", "children" : [ ] }

]
},
},
]
}

On an added note:
update on 'children.$.children' will not add inside

{'$': ...}