Heroes of Might & Magic 3: If you haven't played this game... well, I don't know what to say other than shame on you. Shame! This is one of those few games I find appeals to just about everyone, and it's aged wonderfully to boot. My little teenaged sister, over ten years younger than me who's currently obsessed with anime, even she loves this game. It's just one of those few games that tend to appeal to everyone in one way or another, while also being so very, very good! It's just a prime example of a game which somehow does everything right, and those are surprisingly rare.
Baldur's Gate 1 & 2: Quite often you see people talking about how much they love/hate Bioware basing their entire opinions on recent works the likes of Mass Effect and Dragon Age. That fine, but if you really want to have some perspective on Bioware or even rpg in general, go back to Bioware's first big rpg hit. Take a look at the series that made them who they are and what many people believe to be their magnum opus as a studio!
The Longest Journey: Often people read, watch, or even talk about gaming as a story-telling medium across forums and gaming sites as well as maybe with friends. If you're one of those people, play this game. It is in terms of story-telling
(In my opinion) the proverbial peak of the adventure genre! It told a beautiful, unique story with some of the most compelling characters in gaming today all stunningly brought to life with an amazing script and solid voice acting. The games puzzle elements were lacking, but as an example of story-telling in the gaming medium you honestly can't do much better, and it's a shame the game is so often overlooked.
Plenty of other games I really love or enjoy, but those are the three I think everyone who wants to call themselves a 'gamer' and whatnot should give a shot. Each for their own reasons. All also available at
GOG.com. Those guys oughta be paying me for all the times I've linked that site on these forums. Shesh.