Isn't this the Zerolemon battery on Amazon?
I bought one of these about 6 months ago https://www.amazon.com/dp/B00GYD782A/ref=cm_sw_r_other_awd_TIWMwbW5YX69Q it just about doubles the thickness of the phone and adds a bit of weight, but the trade off is I can go 2-3 days without charging and I am always on my phone.
This is the one I got https://www.amazon.com/gp/product/B00GYD782A/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1#HLCXComparisonWidget_feature_div
It's currently unavailable, but you can see related items which would work as a substitute.
Guy at work has a Note 3 and really likes this case: http://www.amazon.com/i-Blason-Samsung-Holster-Kickstand-Locking/dp/B00F0P2FNU/ref=sr_1_4?ie=UTF8&qid=1389296883&sr=8-4&keywords=note+3+case
I got this for my Note 3: http://www.amazon.com/Zerolemon-Zeroshock-Shockproof-Dustproof-Compatible/dp/B00GYD782A/ref=sr_1_1?ie=UTF8&qid=1389296912&sr=8-1&keywords=note+3+zerolemon
I love my case. Little thick, but with a battery more than three times the size, it's understandable. I feel that the added thickness and weight gives the phone a more proportionate feel. The stock Note 3 is so tall and thin, this extra weight and size makes it feel more natural. To me, at least. And about five days on battery? That's pretty sweet too.
Edit: Found the weight. The phone and case together is 338g, or about 3/4 of a pound. It's also about 3/4 of an inch thick with the case. But it comes with a belt holster. With that on, I don't even feel the weight.