[Galaxy S 4 Q&A, Help & Troubleshooting] abbaskip:
EDIT: My model is I9505

Whilst on holidays, I took my phone out of my bag (when I knew it was getting close to flat) to find it not turning on. I assumed the battery was dead, but when I plugged it into the charger it didn't charge. I fiddled around a bit, and eventually got the vibration and charging light - but it kept turning off before the green battery came on saying charging. Being on holidays I didn't have my USB jig, or even external battery charger with me. I assumed it was a loose connection with the charging port - as every time I tried to plug it into charge it would only sporadically say that it was charging - but never get past the picture of the battery and the swirling icon (prior to the green battery coming up).

Upon arriving home, I got out my external charger, took out the battery, charged it for a few hours, then tried it. Again - it wouldn't turn on.

I then tried plugging the phone in again - but to the 800mA charging doc micro USB, insteead of the standard 2 amp. Straight away I get the charging picture, and the little graphic showing me the phone is about 1% charged. I leave it, the screen goes off. I come back 15mins later, the phone is RIDICULOUSLY hot, and the screen off. I unplug and replug to find the battery is still at about the same 1% level. I turn the phone on - it boots, shows 1/2% and then turns off as soon as I unplug it from the dock - even if I plug it straight into a 2 amp charger. I now decide that maybe the battery itself is the issue - and it's not holding charge. So I whiz out and buy two replacement batteries (MPJ ones, as they were fairly cheap - and this was just for the sake of finding the issue).

I charge one for about an hour on the external charger (the phone didn't turn on when I put the battery straight in). Then put it in - same deal. Nothing. I plug it into the dock micro-USB out to charge - again it comes up as charging, and shows at about 1%. I turn it on, and again am shown 1/2% charge, and again as soon as I unplug from the micro USB dock it turns itself off.

It is almost as if the phone isn't taking any power from batteries, and only direct from the power outlet itself. I'm unsure what could possibly have caused this. I can only think it may have been an external battery which I was using for a while on holidays (which had a 1amp micro USB out, that I was using to try and keep my phone charged). That is the only thing I have done different over this period (at least as far as I can recall!)

So basically - in short:

i) Phone won't turn on unless plugged in to a low amp charger (Computer USB, dock micro USB out etc)

ii) Even after charging batteries in external charger (two different external chargers have been tried) the phone is showing 1/2% battery

iii) The phone turns off a few seconds after I unplug it from the power source.

Any ideas!? I will take the phone to get repaired after Christmas (if they still repair it, as it's running a custom ROM).

BTW. a USB jig worked for the first time this afternoon - when I tried it not long after taking the phone off the dock. It's also worth noting that the phone is ridiculously hot when on the dock (around the camera/sim/micro SD area).