Whoever said that power corrupts obviously didn't own a smartphone. Absolute power is an absolute necessity, all the more so with the big screen, fast processor and multitasking capability of the S4. All of the fun and games you'll be having with the phone will make you an angry bird (lol) if you run out of juice.
For that reason, I highly recommend getting a powerbank - essentially, an external battery in a sealed case. Powerbanks come with a USB cable; you connect it to a power source such as a laptop to charge the powerbank. Some charge instead via flip out blades for plugging into the wall. In either case, you carry the charged up powerbank - and the cable - in your pocket, purse, backpack, briefcase or whatever and use it to charge the phone (using the same USB cable) when you see the battery meter sinking into the sunset.
Powerbanks are rated in mAh (milliamp hours), just like the battery in your cellphone. They have about 70% to 80% efficiency; the rest of the energy is lost to heat. This means, for instance, that a 3600 mAh powerbank wouldn't completely fill your S4's 3600 mAh battery. Even a 4000 mAh unit would fall a bit short (do the multiplications - even 0.80 x 4000 is less than 3600). On the other hand, you'll probably hook up the powerbank before your battery is completely dead.
All units reviewed here include 1 Amp ports for use with smartphones. In addition, the units reviewed that are 5600 mAh and up also include 2.1 A or 2.5 A outputs that can be used with smartphones or tablets.
Powerbanks - which some companies call simply external batteries; there's no standard terminology - have become largely commodified, but not the truly small or large ones. Advertised as the smallest is the Powerocks Tarot, a 1500 mAh baby clad in sleek aluminum that will give your S4's 3600 mAh battery a charge to about 1/3 full if you start from fully discharged. The unit is the height and width of a credit card and about as thick as eight of them. It looks great and slips easily into a pocket.
Both of the above are light enough to carry in a pocket even if you're wearing bball or running shorts.
Want more? There are many options in the midrange, but I like the Limeade Burst L50X2M, a 5000 mAh unit that includes two integral USB cables - USB A (the large size you plug into your laptop to recharge the Limeade) and USB micro B (the one you plug into your phone). This is the one I'm carrying in my front pocket on long days when I can't afford to lose power. It's a great unit.
One caveat: initially, the plugs of those integral cables were almost impossible to unhook from the case (in order to expose the connectors). I actually had to try several tools before I could pop them out of their recessed storage slots. After finally freeing the plugs and sliding them back and forth in their slots, they loosened up, though the micro B connector is still somewhat tight. The company sent a second unit for me and it was fine though. Both came via Amazon, rather than directly from the company, so I think I'm getting a straightforward view of what actual purchaser experience would be. Even with the initial tightness in the one sample, having integral cables is a complete win. So don't be put off by any issue with manufacturing tolerances; just have a screwdrivers or a quarter handy to use for prying out the cables the first time.
For something more colorful, try the Neptor by Eagle, a 5600 mAh puppy available in an array of bright shades.
At 6000 mAh a nice unit is the Fuel+ from Patriot, the memory company. There's a 9000 mAh Fuel+ available too, and several other sizes as well. All sizes at 5200 mAh and above have two output ports, one of which is 2.5A. The company says that's the highest available (other manufacturers offer 2.1A), and that it results in faster charging.
Offering high fashion pour les dames is the Lepow Moonstone 6000. It's shaped like a woman's compact and is available in about eight high-gloss colors. It's a very chic design that would be fine for a night out clubbing.
If you want a unit that charges from an AC outlet rather than from a USB port, go for the Newtrent (aka New Trent) Travelpak+ / NT700C, a 7000 mAh unit. It has integral flip out AC power blades. The unit is significantly thicker and heavier than the Lepow, which perhaps is from the circuitry needed to convert AC to DC.
I'd love to see a power bank that combines flip out power blades with integral cables - i.e., a cross between the New Trent and Limeade devices - but I'm not aware of any. If you are, leave a note in the comments.
Now let's move up to some units that should satisfy the power piggies among you. Bear in mind that the bigger you go, the heavier the battery. But if you have a tablet in addition to a smartphone, you'll need the extra juice. In that case, go for the IOGEAR GMP10K, which despite the 10K in the name is actually an 11,000 mAh power bank. It has a minimalist white look that will complement an iPhone. (Wait, isn't this an S4 oriented review? Yes, but hey, gotta give credit where credit is due.) The others in this review are rubberized black plastic unless otherwise noted.
If you need even more power, there are a few choices, but the one with the most fun form factor may be the Digital Treasures Powerflask with a whopping 13,000 mAh capacity. That's enough to charge the S4 almost three times (allowing for the 70% to 80% efficiency typical of external batteries). It's looks like a flask - complete with faux leather - but only your devices, not you, should be drinking from it. The shape is more than a gimmick though - it means the unit actually fits in your back pocket, making it easier to carry. Don't sit on it though. And don't fill it with gin.
Even thirstier? Try a larger drink of Limeade - the Limeade Blast L180X, to be precise. At a whopping 18,000 mAh, it'll quench your S4's thirst roughly 5 times over. (Alas, no integral cables on this unit.) Limeade also makes two other high capacity units (not tested): the Limeade Blast 156X (15,600 mAh) and Limeade Blast 130X (13,000 mAh).
Next up: the Anker Astro Pro 2, a 20,000 mAh battery, clad in aluminum, big and flat. That's flat, not fat - it's actually no thicker than any of the other units reviewed here except the svelte Tarot. ThAstro Pro 2 can charge laptops (at 12v, 18v or 19v) as well as tablets and smartphones. In fact, it even includes an AC adapter (power brick) that you can use directly with a laptop in case you've lost your power brick. Also in the package are ten different laptop power tips, maximizing the chances that one will fit your computer.
On the subject of laptop chargers, there's also the Targus Laptop Charger (AC) and Backup Battery for Smartphones APM035US. The battery is small - 2000 mAh - but the charger includes a sort of dongle you can optionally use to give you a USG charging port. That means you can charge both your laptop and smartphone from a single outlet - great for use in outlet-challenged hotel rooms. It comes with 7 laptop tips.
That's it for chargers and external batteries. Look for my next installment, where I examine USB chargers (don't yawn, I found some unusual stuff). Afterward will come USB cables (ditto, some cool stuff), video connections for the S4, such as connecting your cellphone to a TV, then keyboards and mice, how to use your cellphone as a portable office, and finally miscellaneous accessories. You can find all this at http://www.huffingtonpost.com/jonathan-handel/.
Disclosure: Manufacturers provided product for this review