A new mod Half-Life 2: Aftermath brings some content from the canceled Half-Life 2: Episode 3 into the game (dsogaming.com).

Recently, there was a lot going on around Half-Life 2: Episode 3. First of all, former game’s writer Marc Laidlaw shared his take on the story of Episode 3. Then a group of developers from Lever Softworks has released a total conversion mod for Half Life 2. This mod adds the content from the canceled Episode 3 to the game. To make it work, you need to have Half-Life 2: Episode One and Half-Life 2: Episode Two to be installed and working. You also need to install Source SDK Base 2013 Singleplayer.
In the mod you will find maps from the canceled Half Life 2: Episode 3. These are not final levels. Level Softworks got access to all the prototype levels that Valve created for Half Life 2: Episode 3. The devs decided to recreate/polish them and release to the public. Overall, the mod contains 11 levels created by Valve between 2012 and 2013. All these levels were intended for Half Life 2: Episode 3. Plus, you also get 4 demonstration levels created by Lever Softworks to show various functions and NPCs not seen in those levels.
There’s a high chance this is the closest you will ever get to the Half-Life 2: Episode 3, so you should definitely try to check it out. Lever Softworks will continue working on the mod, if Valve doesn’t kill it sooner or later.