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

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