Well, never say "never," but it certainly looks unlikely. When it comes down to it, at this point producing Half-Life 3 just isn't profitable enough for Valve to spend the time to do it. On top of the billions from distributing other developer's games via Steam, Valve is also making cash hand-over-fist via transaction fees, in-game items, surcharges and other distribution deals with their Steam Market, subscription plans, non-gaming software, Steam Music, hardware, and even movies.
In Gabe Newell's own words: "The only reason we'd go back and do a super-classic kind of product is if a whole bunch of people internally at Valve said that they wanted to do it and had a reasonable explanation for why [they did], but if you wanna do another Half-Life game, [...] that seems like a bad choice, so we'll keep moving forward, but that doesn't necessarily always mean what people are worried that it might mean."
That being said, with the Source Engine available for game developers to use and Laidlaw's story serving as a basic template, some savvy game developers might decide to do the impossible themselves and bring us a fan-made version of Half-Life 3—even if it couldn't go by that name or use the same characters.
So there you have it—all the reasons we never saw Half-Life 3, and why we likely never will. Of course, this could all change in an instant. Gaben could descend from on high tomorrow and break the internet by tweeting "#hl3c." Maybe this article contains super-secret HL3 numerology and will prompt tons of DMs on Twitter asking for confirmation. TL;DR: the world may never know.