EGPUs to enable utilization of (even) more monitors with MacBook Pro

This answer assumes you're asking about the model still currently available, as the MacBook Air hasn't been upgraded since early 2015 (9 March 2015 to be exact), although there was a very minor spec update in June due to a 12.5% increase in the processor's speed. This, however, has no impact on the integrated GPU being used and even Apple's website does not list this as a "new" model.

Originally this model was only rated by Apple as supporting a single external display of up to 2,560 x 1,600 pixels. However, it was found to actually support Ultra HD (i.e. 4K) resolutions of 3,840 x 2,160 pixels and Apple officially revised their specs.

Most resources (including everymac.com) imply that the MBA only supports a single external display. However, this is not strictly the case, so long as you're using Thunderbolt displays.

According to Apple:

These Thunderbolt-capable Mac computers can support up to two connected Apple Thunderbolt Displays:

• MacBook Pro (2011 and later)
• MacBook Air (Mid 2012) and later
• iMac (2011 and later)
• Mac mini (2011 and later)

Source: About Thunderbolt ports and displays

So, you can use two Thunderbolt displays with a MacBook Air, but you will need to daisy chain them. Note this applies to all models of MBA from Mid-2012 and later. In some cases users have had to do this by, for example, connecting a Thunderbolt display directly to the MBA and then connecting another powered Thunderbolt device (e.g. a Thunderbolt drive) to the display, and then connecting your second Thunderbolt display to the Thunderbolt drive.

Other options

I have seen setups in which users have connected one Thunderbolt display directly to the MBA's Thunderbolt port and a second DVI display via a USB-DVI adapter. Some seem to work quite well, others not so much. If your intention is to just do some word-processing and browsing, this'd be fine, but if you wanted to use the display for more intensive tasks, I would not go this route.

It's a matter of available bandwidth of the Thunderbolt port, not the dock itself.

A Thunderbolt 3 port has enough display bandwidth to handle a single 5K @60H display or two 4K displays at 60HZ.

On the 15" MacBook Pro (2017), it has two Thunderbolt controllers, so you can now attach a second 5K display or two more 4K displays. But because it's on a second controller, it will have to be on a different port.

From Apple's Thunderbolt Page

Thunderbolt 3 provides twice the display bandwidth of Thunderbolt 2, enabling iMac to support two 4K displays or one 5K display. And with two Thunderbolt controllers in the 15-inch MacBook Pro, you can send graphics to dual 5K displays for the perfect high-resolution photo and video editing setup.

The dock can support 2 displays, because that's the maximum that can be supported by the port. To get a 3rd and 4th display, you would have to use a different Thunderbolt port on the MBP.

However, you could (theoretically - meaning I have never personally tested this) put a USB to DVI/DisplayPort/HDMI adapter on one of the USB ports of the dock and get a 3rd display, but I currently can't see how you would get 4K out of it.