I've got a Zerolemon 3000mAh battery for my S4, and it works great. Same size as the OEM 2600mAh, but 15% more capacity. The trade-off is that there's no NFC chip. I've also got a 7800mAh Anker battery/case, but this thing makes the phone a brick. Mainly, I keep it around in case I need a wifi hotspot away from a plug socket for a long time. The 5200 or 5800mAh extended batteries by Anker or Hyperion are basically double-thick, whereas the bigger ones have a much larger battery slapped on top of the regular sized one. Thus, the case thickness will be about the same, but weigh less.
Anker, Hyperion, and Zerolemon all get glowing reviews on Amazon.
I think you either got hosed with used or knockoff batteries, or there's something going on with the phone that's causing excessive drain. Did you do a factory reset when you got the phone? What apps are using all the juice?
This is actually not the case with lithium rechargeables. It was true with NiCads, but lithiums don't need "conditioning" like NiCads did.