I would make it an optional experience.
I've recently began playing a game called "Raid HQ" (no not trying to advertise, they get enough already) and they have a notification for EVERYTHING. Here's a screenshot example of their notifications options:
As you can see, with everything enabled, I could get a lot of notifications. Right now, as I've just begun using it a week ago, I'm still allowing it to scream at me whenever it wants to, and so far I'm happy with it. I might get 10-20 notifications a day, but I like the app, and I want to use the app. So getting alerts for things I want is in my interest. Now if I didn't want to be screamed at for something such as "Battle Alerts" I would just turn it off, yet still enjoy using the app. Right now I just got a "Free card pack is waiting for you" alert. Which I like because it takes 3 hours before I can get another after using it. So in this use case it is in my interest to keep being bothered by it.
However this is for an app which by some definitions, as others have said here, "if it is cool, you can get away with a lot more" (paraphrasing).
If your boss really wants to annoy the crap out of people, just be polite and give those people who don't want to be pestered an "opt-out" option. That way your boss gets off your back, and you, and others, can enjoy a better peace of mind.
And I second/third the idea previously mentioned by someone else to ease up over time if the user isn't using said feature or app. Facebook does this I believe, as I never use it now and as a result instead of getting a notification every time a close friend of mine does something, I will get a monthly notification such as "You have 56 comments, 28 new likes, 8 friend requests and 10 shares" or something like that.
But yeah, add an opt-out option for that specific, and other notifications.
A good general rule is if it's going to spam them outside of the app, let them decide if they want it to stop without having to decide to remove it entirely just so they can get some sleep.