Were I to plug my iPhone’s AC adapter in to the outlet of my UPS, would the UPS system recharge my iPhone (regardless if the UPS were in AC or battery mode)?

Or would chaos ensue?

(While it seems [to me] that it should be ok, what is intuitive to me, ain't always so.)

In this circumstance, your intuition appears to be correct......the output of your UPS should drive the iPhone AC adaptor just fine regardless of the UPS battery -vs- mains issue.

Many thanks, kind sir. Your prompt reply is appreciated.

And to beg a bit more of your expertise...

As the battery in the UPS is of much greater capacity than the one in my phone, would the phone (upon connection) eventually be fully recharged, even if the UPS is in battery mode?

Assuming you plug the iPhone charger into a 120 V AC outlet there would be no different than plugging the same device into the wall. 120V 60 Cycle power is 120 V 60 cycle power is the same whether it comes from the wall or the inverter in the UPS. Assuming there is no other device pulling power from the UPS it should completely charge your iPhone just as if the iPhone were plugged into the wall socket. Assuming of course the UPS battery is fully charged and there is sufficient battery capacity to provide 120 V AC power for up to 8 hours. There is always some power lost to heat when inverting 6, 12, or 24 volt DC to 120 V AC.

And thanks to you, joemike.

I don't foresee that I would ever need that capability, but 'tis good to know, and should I happen upon a luckless soul in such a (dead battery) quandary, perhaps your guidance will save the day.