Are pop-ups ever not annoying? :)
Been on mind a lot lately actually, and I basically cannot come up with a situation where popups are actually an unavoidable and proper good choice. Not from a user perspective anyways (from a dev perspective, it's an easy way out and a "good" way to attention grab... and then not hold).
Sometimes it grabs focus from whatever I'm doing in Illustrator and this is indeed slightly annoying but it is also useful since I want it to interrupt whatever I'm doing and make me ask myself if I am at a good place to save, and if I'm not, then to save as soon as I am.
Arguably this is still annoying but it is an annoyance I have explicitly asked for, knowing it'll be annoying.
Pressing space to "preview" a file in Finder on macOS is pretty much "non-annoying popup", since you actually want it :)
Some disappear if left unclicked, I guess if the problem is no longer relevant (e.g. a patient bored of waiting leaves), and they can be attended to in any order.
How about pop-up with 2fa request? Better to have pop-up for push notification than to look for auth app by yourself.
Personally, I wouldn't call this a popup because this was a result of a user action.