Dragon Age: Inquisition now has a demo, sort of. It's actually a "limited time only" trial—with no word on how long it'll be available. If you haven't bought the massive Bioware RPG, though, it's well worth checking out. You'll get six hours with the singleplayer portion, and unlimited free access to the (admittedly not great) multiplayer mode.
The multiplayer works a bit like Mass Effect 3's excellent horde mode, only—due to the fact that DA:I's combat is one of its weakest elements—its nowhere near as good. Still, it's received a number of updates since I last played it, and, of course, it's free.
More useful, I suspect, is the chance to try out six whole hours of the campaign. That's about 5-10% of its full length, depending on your desire to complete sidequests. My advice? Don't hang about in the Hinterlands. Six hours won't get you to the best bits of the campaign, but you can still see a lot as long as you're not distracted by its starter zone.
To download the trial, head over to this Origin page.