MongoDB: Too many positional (i.e. '$') elements found in path

You CAN do this, you just need Mongo 3.6! Instead of redesigning your database, you could use the Array Filters feature in Mongo 3.6, which can be found here:

https://thecodebarbarian.com/a-nodejs-perspective-on-mongodb-36-array-filters

The beauty of this is that you can bind all matches in an array to a variable, and then reference that variable later. Here is the prime example from the link above:

Use arrayFilters.

MongoDB 3.5.12 extends all update modifiers to apply to all array elements or all array elements that match a predicate, specified in a new update option arrayFilters. This syntax also supports nested array elements.

Let us assume a scenario-

"access": {
    "projects": [{
        "projectId": ObjectId(...),
        "milestones": [{
            "milestoneId": ObjectId(...),
            "pulses": [{
                "pulseId": ObjectId(...)
            }]
        }]
    }]
}

Now if you want to add a pulse to a milestone which exists inside a project

db.users.updateOne({
    "_id": ObjectId(userId)
}, {
    "$push": {
        "access.projects.$[i].milestones.$[j].pulses": ObjectId(pulseId)
    }
}, {
    arrayFilters: [{
        "i.projectId": ObjectId(projectId)
    }, {
        "j.milestoneId": ObjectId(milestoneId)
    }]
})

For PyMongo, use arrayFilters like this-

db.users.update_one({
    "_id": ObjectId(userId)
}, {
    "$push": {
        "access.projects.$[i].milestones.$[j].pulses": ObjectId(pulseId)
    }
}, array_filters = [{
        "i.projectId": ObjectId(projectId)
}, {
        "j.milestoneId": ObjectId(milestoneId)
}])

Also,

Each array filter must be a predicate over a document with a single field name. Each array filter must be used in the update expression, and each array filter identifier $[] must have a corresponding array filter. must begin with a lowercase letter and not contain any special characters. There must not be two array filters with the same field name.

https://jira.mongodb.org/browse/SERVER-831