You must compare watt-hours (or use joules, if you like) for it to be meaningful. AAA voltage is 1.5V, lithium ion voltage is 3.7V or 4.2V or something, which means that each "mAh" is worth several times as much as one from a AAA battery.
The iPhone 14 Pro battery is about 12Wh. A single AAA battery has about 1.9Wh, according to google. So, 6 AAA batteries, optimistically, excluding other factors that could raise the count. Using your number of 800mAh*1.5V=1.2Wh, which would mean 10 AAA batteries.