How to run 4 external displays?

You are running into a limitation of the hardware.
You have three video device outputs, 2 on thunderbolt, and one HDMI.
You can chain two thunderbolt devices (or one TB in each port) and add the third to HDMI, then you get no video from a 4th device plugged into the 2nd TB port, or add a third display to the 2nd TB port first, then HDMI won't light up a 4th display.
Adding a 4th device will remain black. You may be able to count the internal display as the #4 display (I think), but 4 externals won't work.
(Wait for it - someone will prove me wrong with a picture of 4 external displays running on a rMBP)

 

Yes, but notice that El Gato takes the place of using a daisy chain connection, when the display the you want to use has none.

Anyway, your MBPro has two TB connections, so you can connect two TB displays. Note that you could connect the two displays to a single TB port, assuming the display has a second TB port to allow daisy-chaining.
If you do THAT, you can (I think) attach a third display to the MBPro second port.
No daisy chaining on THAT display to add a 4th display, even if it has the extra port.
It's a 4th display, and the hardware won't support it.

Use both TB ports, one display each, add an HDMI display. Works.
Daisy chain two displays from one TB port, add HDMI, Works. Add another display to 2TB port. Won't work.
You have 3 external video outputs, which can be configured in several different ways, maximum of 3 displays.
You can have two attached as daisy-chain, then add the El Gato hub, and use THAT with a single display.
Your MBPro would then have a total of three displays. The El Gato has capability of two displays, but the Mac won't supply video to the second display if you already have 2 displays in use (the thunderbolt bus is supplying all the video outputs possible) Add a third through HDMI (on the MBPro, not the El Gato)
Get the picture? No 4th external display on your MacBook Pro, even with accessory docks.

 

To run four external monitors with a MBP:

1. Go to the Apple Store.
2. Buy a 2016 15" MBP.
3. Connect a monitor to each port. Or daisy-chain.

 

No connundrum the MacBook Pro you have supports a maximum of three external displays. its a limitation of the hardware, you are trying to do something the computer just won't do.

 

OP, agreed with others here about the limit of number of displays. The rMBP doesn't have enough graphics power to drive all of those displays. Additionally, the limitation of TB relative to throughput - TB cables and adapters with TB cabling, by design are limited to DP 1.1a - we never use TB cables to drive displays in my offices, only DP 1.2 cables that are 2160p-rated. Introducing TB cabling into any display-related chain IMO will only gimp the performance of any display; please refer to the DisplayPort Wiki for more on this. Devices like Matrox's desktop extenders don't work (according to Matrox) with the newer rMBPs that have AMD dGPUs, only the iGPU and the results are limited to 768p (ugh).

The only solution I can think of is to look into a eGPU set up, which I do employ for some of my Windows laptops in one of my offices but not for any of my Macs. eGPUs for Macs aren't on my radar and there's others in these Forums that have the capability to advise you about these graphics boxes. Good luck.

 

With 2015 15" You can run at least 5 displays simultaneously:
1: Internal
2: Display Port
3: Display Port
4: HDMI
5: AppleTV

I haven't tried it, but I guess you can also connect your iPad through "Duet" as external display, which would make it 6.
You can also try this adapter, should add another monitor:
https://www.amazon.com/gp/product/B00BPEV1XK?psc=1&redirect=true&ref_=oh_aui_detailpage_o05_s00

 

Ah, good catch! I didn't think about the AppleTV.

The OP wanted 4 external displays simultaneously, so I guess that does it with the AppleTV
Is there any lag on the video through the AppleTV (compared to the other displays?)

The USB-HDMI adapter that you linked is not compatible with Sierra.

 

There is a lag, of course , but it looks usable for text , not sure how the op wants to use it. And I guess you don't have to run Sierra if your use case is not supported, but i don't think the adapter will be more usable than Apple TV anyway.