You're not going to get a battery that has more mah AND fits under the stock battery door. Some batteries state a higher mah, but have the same or less than the stock battery.
I agree with
@rbiter, spare Samsung battery with Samsung battery charger = awesome combo! That's what I have and love it! I also have a 12,000mah power bank with 2 usb ports, so it can charge 2 devices simultaneously. This combination gives me more than enough battery life to last me for days without needing AC or DC power!
SwiftKey'ed from my White Sprint Note 3 via XDA Premium