If you charge a lithium battery at faster than 1C (IOW, if it's a 7,000 mAh [7 AmpHour] battery and you charge it at more than 7,000 mA), you'll make sure it has a short life. (Which is why a 5 minute full charge may be great marketing, but it's lousy engineering.) Charging a lithium battery too fast can cause it to explode (from gas pressure build-up) or catch fire (from overheating).
I prefer carrying a spare battery or two, and have a charger to charge them outside the phone. They charge at the normal rate, I can swap batteries at 40-50% (which gives the longest life) and it's cheap. (For most phones, 2 good batteries and a charger run around $25.)
I saw a 4500 mAh battery on eBay that claims to be the same size as ours. Anybody ever try it it's only $7 so I'm worried if it's any good
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If you haven't seen an extended battery that fits within the original case - you'll have to get out more. By "extended" I mean a higher mah level than stock which is 3220. I used a 4200 mah on my N2 to great advantage. I now own 2 extended batteries one rated at 4500. I am trying to avoid those thick heavy batteries as good as they are. Unless you're going on a long expedition with no access to electric, or solar power I don't see a widespread need for 10,000 mah. Those within the case extended batteries are widely available on Amazon and EBay , although I've had some bad experiences with vendors from our friends in China.Haven't seen an extended battery that fits inside the original case yet. I ordered ZeroLemon » Zerolemon Samsung Galaxy Note 4 10000mah Extended Battery + NFC + Rugged Black Zero Shock Hybrid Protection Case – Black for myself based on past experience with S3 and S5 kits from them. They include a quality case with the battery, BUT be warned that the phone will be nearly 21mm thick with this setup and as such is not for everybody. Full NFC support for S-Beam, Google Wallet, etc.