There are probably a few dozen reasons why Half-Life 3 has never surfaced, but fans of the series have set their sights on another Valve game for being the primary culprit: DOTA 2.
If you happen to peruse the DOTA 2 review scores on Steam, you'll notice a rather impressive influx of negative takes on the popular MOBA, despite the fact that the game has been out a few years now, it has been constantly supported and fans seem pretty happy with the way things are going.
It seems like Half-Life fans have decided that Valve piling all of its resources into DOTA 2 has now been blamed as the primary reason a proper follow-up has never been developed. You can tell because none of the reviews have a thing to do with the game they are actually reviewing. Instead, we find comments like "Wheres HL3?", "I need my HL3 no more steam purchases," and "It's not half life. It's not original. Why, valve?" I quoted those directly, so don't blame me for the butchering of the English language.
Few vaporware stories stir up as much salt as the lack of a continuation to the Half-Life series. The first game came out at the turn of the century and basically served as the blueprint for the modern first-person shooter. Half-Life 2 was just as stellar and boasted a follow-up "Episode 2" to continue its story. That second chapter ended in a cliffhanger that was supposed to be concluded with a sequel or, per the original plan, "Episode 3."
That, of course, never happened, despite the fact that rumors about it "finally happening" pop up every six months or so.
To be clear, the Half-Life series went cold long before DOTA 2 came out, but it seems like fans are upset that the game's success (and draw on resources to keep it running and relevant) appears to be taking all of Valve's attention. In other words, they think that if DOTA 2 didn't exist, they'd probably be playing Half-Life 3 by now. That doesn't exactly mesh with the way Valve is known to handle development, but I suppose folks are free to believe whatever they like.
The unfortunate thing is that these folks are responding in an increasingly popular manner: infantile down-reviewing. If they can't get the game they want -- the game they seem to believe they are entitled to -- then they're going to try and make Valve feel some pressure. Never mind the fact that reviews probably aren't affecting a years-old game at this point and that said reviews are on a site Valve runs and can scrub clean when needed.
This all seems to have been spurred on by a recent blog post from a former Valve writer, Marc Laidlaw. Over the weekend he laid out a fictional tale that was about as transparent as you can get, highlighting what pretty much everyone has accepted would have been the continuation of the Half-Life series. Not that this means Half-Life 3 is never coming or that it will even follow Laidlaw's outline, but folks have taken it as some sort of proof that Valve has put the kibosh on the series and, um, DOTA 2 is responsible? Not Hearthstone, though. That game is apparently free and clear of blame.