I cannot be entire sure about this but it seemed to happen this way:
When the iPhone 5, 6, or 7 was out of power, I could plug it into my car's USB charger or a portable USB battery charger for 10, even 15 minutes, before the phone is willing to start up. A 15 minute wait, can be a bit too much -- if it were 2, 3 minutes, it would be more tolerable, but a good 15 minute, is too much.
But I noticed sometimes when I plug in the lightning jack for a charge, the iPhone will start immediately.
Is it due to some charger are supplying 5W, while some are supplying 10 or 12W?
I suspect iPhone might be doing some calculation: if possible power consumption is less than power input (10W), then be worry free and power up immediately. However, if possible power consumption is more than the input (5W), then don't power up first, accumulate first before powering up (and it turned out to be 15 minutes usually).
Does someone know for sure how this works?
P.S. Decemeber 2016: So today, I tried out an iPhone 5S, and saw that it was 1%, but too late, about 1-2 seconds after I plugged it in, it turned itself off. And thinking that it was a 39W 2 port charger with QC 3.0 (an Anker charger used in the car), I thought it should start up almost immediately. Not so, it took close to 15 minutes before it turned itself on, and I didn't look what battery level it was when it started up, but 10 minutes or so after the start up, I looked, and it was at 21%. So it looked like it could have turned itself on sooner.