I get the same reaction to topics like these! ^_^
I've been playing video games since before I can remember... There are actually photos on me playing some game using an ancient controller (an old Atari console I think) that I don't even remember. I have too many games that I love to mention here so I'll pick the one's that have had some staying power with me over the years and other noteworthy examples.
UFO: Enemy Unknown - The original and classic. The best strategy game ever concieved. For anything. I love the new XCOM game, but I still return to the original now and again because it's so much more open-ended. I actually thought for a while that I was going to make a comic/narrative based on the xcom mythos, or at least a tribute. I ended up making Voidchild instead though.
Xenogears - alright, this one is a bit weird. Because I've gone back to the original game in the xeno-series and I'm not sure it holds up that well today, gameplay-wise. The story is stlll the most anime'ish story ever concocted. I was never a big fan of the Xenosaga-games for gameplay reasons. Then came...
Xenoblade and Xenoblade Chronicles X - The most modern takes on JRPG's, a genre that's hurt me too many times over the years and left me scarred and with trust issues. Finally, someone(Tetsuya Takahasi) sat down and sorted out all the ugly draconic errors that kept repeating themselves in JRPG's since the Playstation One-era.
City of Heroes/Villains (MMORPG) - This is the saddest chapter in my personal gaming history. The penultimate superhero-/villain mmorpg. Create your hero or villain and go nuts. So many powers/builds to choose from and the most robust character creation system to date. Then an evil grubby company (read NCSoft) went and cancelled it. Appearances suggest that they hadn't even notified the developers they were doing this since the devs were hard at work on new content for the game and a developing epic storyline which never got finished because of this. I'll always have the screenshots though.
Super Smash Bros-series - I didn't get into this series until the Gamecube-era. But at that point I was immediately hooked. It's probably the series that I've sunk the most hours into. It might not be the tightest fighing game out there, but it's not supposed to. It demands a degree of skillful play no doubt, but it's mostly just wonderful chaotic fun.
At the moment I'm playing Dark Souls 3 and Doom when I can find the time between making Voidchild and other stuff. Both real good games that I might not recommend for just anyone because they're particular. But I'll mention them here anyway.
Aaand I'll leave it at that so this doesn't go on for too long....