This app was mentioned in 20 comments, with an average of 1.30 upvotes
Prof Reminder is my preferred app for repeating notifications. Free version might not do what you want but the upgrade was well worth it.
I use one called Missed Call Reminder aka Prof Reminder. I have the pro version, so I'm not sure which features are not in the free version.
What I like is being able to set repeat notifications, in case I don't hear the first (or second, third, etc) sound for some reason.
This app allows you to set the sound for different types of notification, and a custom repeat rate with delay for first one. I think it's the pro version that allows it to be set for any app that has a notification
I'm not sure about battery full notification, because this is not something I feel is useful. Modern phones handle battery charging just fine, and worrying about immediately disconnecting the charger when full is, in my opinion, of infinitesimal value, if any. Life's too short to worry about things like that, but then again, I don't have any low-quality phones that might not handle this condition correctly.
This will probably sound like a promotion or something, but thought I'd mention an app I've used for years that I find really handy for notifications. I use it in a way that is different than my original intention, and also from its description.
It's called Prof Reminder, and is advertised as a way to help eliminate missed notifications. The paid version allows you to set customized behavior for notifications for all apps, either individually or in groups. I now use it for all my important notifications. I set all others to silent.
It can set the ringtone, loudness, vibrate pattern, LED, and you can set an amount of time that it will repeat, how often it repeats, ...
So, I have vibration patterns for several reminders, a faint beep with a single vibration pulse for others, and for SMS I have it set to a loud repeating setting so I will be sure to hear it.
It also has its own DND type settings, in case you really want quiet.
You can also look at Prof Reminder as an alternative. It not only does this (paid version only, perhaps), but has a bunch of other options for repeating certain notifications, using sound, vibration patterns, LED. I've been using it for years to repeat message and call notifications in case I miss the first one, and now I usually set all my notifications to silent and use this app as a central place to customize them.
This is what I use. It plays a loud beeper until I acknowledge it.
When I get certain notifications it will make the phone ring until I acknowledge it with no timing out or anything. Has scheduling features and can bypass DND mode. Pretty good if you are a heavy sleeper in an on-call rotation.
I believe there are various apps that can do that. I have one that has that as an option, I believe, although I don't use it for that. It's called Prof Reminder; the paid version allows you to have it work with all apps, and it can do things like repeated notifications, LED flashing, etc which is what I use it for, but it seems that one option is to turn on the screen. There may be similar apps that are more suited to just what you want, though.
I used Prof Reminder. You can setup schedules, etc. I have it set to play loud noises if I miss a call until I acknowledge it.
It also has support for text messages, but thats not relevant to me.
edit: https://play.google.com/store/apps/details?id=com.dan_ru.ProfReminder&hl=en
Missed call and SMS / MMS reminder with profiles. Flash on call. https://play.google.com/store/apps/details?id=com.dan_ru.ProfReminder
I use Prof Reminder for that purpose.