appliedAnnotations_ is private, mangled and actually not even available in production build:
https://github.com/mobxjs/mobx/blob/78d1aa2362b4dc5d521518688d6ac7e2d4f7ad3a/packages/mobx/src/types/observableobject.ts#L114-L117

Do you know if there is a way to get to the annotations in production build? Or is this idea just not possible?

It seems to me that you've misunderstood how ref and shallow works. They say how a non-observable value (plain object/array/set/map) should be converted into observable - if the value is already observable, these decorators have no effect. So if you have an observable Entity it remains observable even if you assign it to ref prop or push to shallow array. Since these don't remove observability, it would be weird if deepObserve would ignore it.

You could additionally wrap the Entity into something actually non-observable to stop the recursion:

@observable.ref _entity = { ref: entity }

get entity() {
  return this._entity.ref;
}

set entity(entity) {
  this._entity = { ref: entity }
}

get to the annotations in production build

I think you can get the enhancer via getAtom(thing, prop?).enhancer, but to find out what kind of enhancer it is, you actually have to call it and check the result, like this:
#170 (comment)

Well, that's exactly what we want. We have entities which reference each other (decorated with observable.ref) and each one have their own observable properties. We want to observe the entities in a way that we get info only when their own property changes, not property of some other entity then reference.

class EntityA {
    @observable.ref entityB?: EntityB
    @observable propertyA: any
    @observable somethingDeep: string[] = []

    constructor() {
        makeObservable(this)
    }
}

class EntityB {
    @observable propertyB: string

    constructor() {
        makeObservable(this)
    }
}

const entityA = new EntityA()
const entityB = new EntityB()
entityA.entityB = entityB

const dispose = deepObserve(entityA, console.log)

entityA.propertyA = 'something' // I want this to be logged
entityA.somethingDeep.push('something') // I want this to be logged
entityB.propertyB = 'something' // I don't want this to be logged

dispose()

In that case you should use observe, not deepObserve. As @urugator noted ref/shallow modifiers are completely unrelated to deep resp shallow observing. Simply put, deepObserve is a wrapper around observe, that sets ups new observe listeners recursively, until no observables could be reached anymore. So if an an observable.ref points to an observable, it will recurse into it. MobX-state-tree has a semantic distinction between 'owning' an object, and loosely referring to it, but in MobX we make no such distinction.

But with observe I wouldn't get the update when doing entityA.somethingDeep.push('something'). Anyway it looks like it is not possible to do this in the production build or you just don't want this to be included.

Closing PR