While there is no official confirmation on this from Valve, it seems they have filed for a trademark for Half-Life 3 on the European Union's trademark and design registry. Additionally, an observant NeoGaf user found a reference to it in Valve's bug tracking software.
The listing on the Office for Harmonization in the Internet Market, the Eu's trademark registry shows that the filing belongs to Valve and is for computer game software, as you can see in the picture below. The filing was, however, filed on Valve's behalf by Casalonga & Associés, a patent and trademark firm. You can jump to the source link and search on "half-life" to see the filing in its entirety. The US Patent and Trademark Office (USPTO) does not contain a filing for Half-Life 3 at this time.
The snooping around of the NeoGaf user shows the game in the Valve's bug tracking lists. The bug tracker site has since been taken down, perhaps in an attempt to hide the evidence, but if it's real the cat's already out of the bag.
So, do think Valve is going to be making a huge announcement in the near future or is it just more wishful thinking by all the gamers who have been waiting so long for the next installment in the Half-Life series? Post your comments.