I have background refresh disabled for most apps. My exceptions are:
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My credit card mobile apps (and with background location services enabled).
As you may be aware, cards red-flag card-present transactions that are far from your normal home/work locations. When traveling, you can call them (or use their web site) to let them know where you will be, to avoid problems. For some cards (including a few of mine), they will use your phone’s location (as reported by the app) as another factor - if the phone is physically near the location of the transaction, they will treat it as if the location was registered as a travel location.
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Notes. If I edit notes on another device (phone, iPod, Mac), I want the content to be immediately synced to other devices. I don’t want to need to launch the app to get the changes, since I might not have Internet connectivity at that time.
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Books. For similar reasons. I frequently add/remove PDFs on my Mac. I want them to sync ASAP.
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Siri. I’m not sure if it strictly needs this, but I have it enabled in case it’s needed for “Hey Siri” to work while the phone is locked.