[Motorola Droid 4] Septfox:
The following is some testing I did out of curiosity. My methodology is decidedly short of scientific, don't take any of it as the gospel truth/hard fact. Proving me wrong is encouraged.

It's also well into reasonably-TL;DR territory. There's a summary at the bottom.

A while back, I bought a Xiaomi power bank, and with it a reasonably powerful charger to fuel it.

During one particularly sleepness night, I'd settled down in bed with my phone - which was nearly dead after laying neglected on my desk for a couple of days - and set it charging whilst reading news and whatnot. A half hour later, I noticed that it had accumulated a surprising amount of charge, despite me using it.

Curiosity piqued, I bought a Charger Doctor sort of thing - which from here on out I'll refer to as a "monitor". It's one of these specifically (chosen based on review found here), to see how much amperage the phone actually pulls if given as much juice as it wants.

Unfortunately, these tests are a bit flawed - I have a pair of 24/28AWG micro-USB cables on order, but for the moment all I have is the input pigtail that's on the monitor (it also has a micro-USB port for input) and a short cable that came with my power bank to go between monitor and phone. I think that the input pigtail is limiting the maximum delivered amperage, because the amperage doesn't rise above the maximum observed even while in use on the charger (more on this later). I'll get another batch of samples when the new cables arrive.

Does not seem to be the case, or at least, the pigtail is reasonably capable. Switching to the micro-USB port actually lowered the current delivered.

It looks like short of soldering a new pigtail in (using thicker wires), there's nothing I can really do to eliminate this potential bottleneck. The pigtail is short enough anyway that it might not be adding enough resistance to make an appreciable difference.

The charger used was a Xiaomi MDY-30-EC (2A output max, Quick Charge 2.0-able apparently). In hindsight, I should have gotten an MDY-30-EH for future-proofing (2.5A/QC 3.0), but you know what they say about hindsight.

I drained the phone to 10% and set it charging, checking on it every now and then to see what it was doing. Then I proceeded to forget about it for a bit, so it kind of got away from me, but the power curve for 10%-33% proved to be fairly flat.

10% ~ 33% - ~1.58a

33% - 1.5a update: observed to still be holding 1.58a up to at least 43% in the second run

50% - 1.14a

57% - 1.05a

60% - 0.965a

79% - 0.865a

80% - 0.775a

83% - 0.645a

85% - 0.565a (0.50a)

87% - 0.505a (0.465a)

87% - 0.475a

95% - 0.165a (~0.115a)

100% - ~0.07a (intermittent)

Note that for much of the time above 60%, I had the screen on at the lock screen and auto-dimmed ("keep awake while charging" on via Development Options). It wasn't until 85% or so that I thought to check if the phone will pull additional needed power directly, instead of draining the battery. It does seem to, so at 85% and 87% I shut the screen off and recorded the draw.

As you can see, the phone will use up to 1.58 amps if given as much, and stays above 1A until 60% (where it suddenly drops to 0.96A). It's only when it reaches about 80% that the stock "fast"/travel charger becomes adequate, and 85% when a standard charger or USB connection can charge it with the screen off, and neither of those leave much if any excess capacity for the phone to consume if it's awake/being used.

Overall, it seems like quite a sane charging curve, edging into overly cautious. I'd guess this is because the battery is technically non-removable, and they didn't want people sending their phones in under warranty for the battery replacements that a more aggressive charging curve might have caused. Not that any of it mattered, what with them only ever offering a 750mA charger anyway.

Again, this test is flawed from the outset due to lack of a decent input cable. I seriously doubt it pulls much more for charging, but a proper cable might allow the phone to pull more for active use; at 10%, the current didn't go above 1.58A even while messing with it.

Fast-charging the battery might also cause longevity issues, because of potential heat build-up. Don't be an idiot like me and set it next to your pillow, only to find it under said pillow or blankets in the morning all toasty and only half charged because the safeties - yes, surprisingly we do have over-temperature safeties - kicked the charging circuit off. Also probably not the best idea to do gaming or anything else intensive on it while charging at full tilt, what with the processor's EMI shield being in direct contact with the battery and all.

Further on this: The heat build-up doesn't seem to be bad, with the phone sitting back-down on a wooden desk. In fact, the casing on the charger itself got much warmer than the back cover of the phone after half an hour of running at maximum input, to the point I started worrying about it instead.

I would tentatively say that "fast-charging" won't hurt the battery, at least, not by way of overheating it.

Some further power draw tests. Anything under .5a might be (probably is) inaccurate to some degree, and the monitor's minimum seems to be 0.07a.

Booting the phone - 0.44a (note that this isn't even enough to power the boot process what with the CPU saturation, so rebooting at <10% is probably a bad idea)

100%, screen on at launcher, brightness maximum - ~0.195a

100%, screen on at launcher, brightness minimum - below cutoff

Following is at 100%, brightness maximum:

GPS on (no satellites in view) - 0.30a

Bluetooth on (active scan) - ~0.42a

Bluetooth on (passive, unlinked) - 0.215a

Bluetooth on (actively broadcasting) - intermittent bumps to ~0.24a

CPU loaded (1.3ghz) (Passmark integer math) - ~0.78a

GPU loaded (Passmark 3d test - complex) - ~0.59a

GPU loaded (3dMark Ice Storm) - ~0.42a

3dMark physics test - ~0.68a

In summary:

We can use up to 2A chargers (possibly more if excess capacity is needed by the phone being under load while charging), despite only ever having been officially offered a 750mA charger at most. Thanks for that, Motorola.

The charging circuit supports bypassing the battery for direct power, if charger capacity beyond what's needed for topping the battery up is available

The screen uses a fairly tiny amount of power at minimum brightness

The screen uses a lot of power at max brightness

The GPS also uses a lot of power, roughly half that of the screen at full brightness if measurements are accurate

The GPU uses a large amount of power (note: real-world usage for the GPU outside of games is normally very bursty, but tends to get saturated a lot for UI acceleration in LP/MM)

The CPU uses a ton of power (note: real-world usage generally has the CPU napping at 300mhz where it barely consumes anything, and/or bouncing between 600mhz-1ghz)

More:

The charging circuitry seems to be able to utilize capacity above 1.585a, based on momentary spikes to 1.6a/1.64a. I'd guess my charger is a limiting factor here.

Our phone meets and exceeds Quick Charge 1.0's nebulous "30% in 30 minutes" marketing blurb/guideline.