Can I use two non-Thunderbolt displays on a MacBook Air I have been told that the only way to connect more than one external display to a MacBook Air was to buy Apple's Thunderbolt monitor and daisy chain.

Mavericks promises better support for external displays. Is it now possible to run two non-Thunderbolt monitors from a mid-2013 MacBook Air or is this still impossible?
The changes in Mavericks make it possible to connect a DisplayPort monitor to your MBA and also use an Apple TV as an external display - I've just confirmed it on my setup - but that is pretty slow over wifi. 

The issues with connecting monitors to the MBA are in hardware, not software. For example, the reason why you cannot connect a DisplayPort adapter to the Apple Thunderbolt Display is because the monitor itself uses the DisplayPort channel:

http://www.anandtech.com/show/4832/the-apple-thunderbolt-display-review/8

Connecting two displays to the MBA requires two thunderbolt devices.

The first device can be either the Apple TB display or one of the Thunderbolt docks with a DVI / HDMI connector. Both of these options use the DisplayPort channel, so connecting a DisplayPort monitor to them won't work.

This is where the second Thunderbolt device comes in (a hard disk or other device with two TB ports for pass-through). This acts as a man-in-the-middle and can extract a second DisplayPort signal.

So your setup will look like this:

&lt;Air&gt; -- &lt;TB dock/display&gt; -- &lt;TB drive&gt; -- &lt;DisplayPort monitor&gt;