I mean a game you play fairly often and have been for a long time. Maybe you’re an ardent chess player and have kept it up for decades, or there’s a computer game you love so much you keep playing it despite it being hopelessly outdated, as far as the game industry considers such things.
Mine is Wordfeud, one of the top Scrabble clone apps. According to the in-game statistics I’ve been playing for 10 years, over 1800 games. 958 of them against my dad, who’s in the lead 611 to 340.
Dungeons & Dragons – I’ve played five different editions of the game, and have been playing it regularly since 1982. In fact, I have an online session of a 5th Edition game coming up later today. 
When Skyrim released its anniversary edition, I upgraded and started a campaign. I’ve been playing that game off and on for ten years.
I should note that I also occasionally play Strat-o-Matic Baseball and Strat-o-Matic Football; I’ve been playing both games since sometime in the early-to-mid '80s.
Same. I’m doing a Session 0 for my new campaign in about four hours.
Cribbage, I’ve been regularly playing it forbabout 40 years.
D&D since 1977. Before 1st Ed. I have a beat to crap 1st printing of the original DMG in fact from the day it was available for sale. Exciting stuff for us nerds at the time. I used and abused that book until about 5 years ago.
I’m now on 5e, having largely skipped everything in between.
I’ve been playing and modding Age of Wonders Shadow Magic since around 2004. A great fantasy turn based strategy game.
Traveller I still play occasionally. Started in 1980.
Battle Tech also still play occasionally, Started around 1986.
Last Chess game was 4 years ago, so probably doesn’t count, but started when I was 4 around 1970. Never got that good, but in 1984 we won the HS Nationals. I was 2nd board.
My siblings and I grew up playing Rook. I still play occasionally when the opportunity arises. That would be 60+ years.
Another WordFeud fan here, have been playing that pretty consistently for 10+ years as well. Otherwise our family’s go-to game is a card game we call “Bump” but I’ve seen called “Oh Hell” in other contexts. I learned as a child with my father and have been playing it on and off with family for 50+ years.
Like many others, D&D, since the early 80s. I’m not entirely sure that counts by the OP’s standards, since I play the latest version, “5th Edition”*, which is not outdated. There are folks who still play some variation of the AD&D, B/X, BECMI versions that were current 40 years ago.
Also seems like it doesn’t quite count since I always play the current version, but while I don’t play many video games, I do play a lot of Madden Football, and have for a couple of decades now.
*Officially, the current version is simply “Dungeons & Dragons”, and the 5th Edition moniker is a fan convention to distinguish it from earlier variants. Also, depending on how you count, it’s really at least the sixth edition, and maybe more like 10th Edition.
BattleTech through various platforms since '91.
Starfleet Battles through various platforms since '95.
Tripler
I never could find a D&D group to join and learn from. 
The “outdated” comment was specifically for computer games. I don’t think D&D or Wordfeud are outdated.
I’ve played D&D for the last two years, though the current campaign I’m DM-ing is on hiatus for scheduling reasons. But I’ve not been RPG-ing regularly and my previous forays, in my teens and at university were all in other systems. Did the Scandinavian D&D “rip-off” Drakar og Demoner (Dragons and Demons) as a teen and Earthdawn, Stormbringer and Paranoia while at university. Our GM wasn’t much for settling on one system and we were all just playing like munchkins.
So all of that paragraph is off topic for the thread!
Sid Meier’s Civilization series for 31 years (I’m currently on Civ 5.)
Dungeons and Dragons for 43 years (and our group is still on 1st Edition 
)
Bridge for 52 years.
Chess for 62 years (taught myself from ‘Chess for Children’ when I was 6 - and never looked back. 
Tetris, since whenever it debuted in the US (mid-1980s if memory serves).
Resident Evil 4 certainly meets the definition of “hopelessly outdated”, but I find myself revisiting every few years and enjoying the hell out of it.
I’ve been playing Black Knight pinball since the early 80’s (except when it is broke, which is most of the time). Hell, I even play it when it’s broke…
I started playing this Stoopid Forge Of Empires a few years ago. Regret every single day of it. Can’t post anymore, got to go tend to my stoopid world and it’s stoopid challenges…
I played ABPA Baseball starting at age 11 so 62 years ago. I don’t play it so much anymore. I first learned bridge as a teen and played it regularly until 2 years ago. I don’t see well enough at night to drive to the evening games, but I do play some online. I started playing cribbage and backgammon when I met my wife so 50+ years ago. I still play cribbage with my wife and son and some backgammon.
Computer:
Flight Simulator (I started with the Sublogic monochrome version from the mid-1980s.
Civilization (I have owned every version, including VI, but prefer V)
World of Warcraft (I bought in the first couple of months it came out; I think I’m probably over it now; I was super-excited for Classic, and played for a while, but playing Elder Scrolls Online made me realize just how lazy the WoW questing is)
Physical:
Risk (I was introduced to it by my best friend Dwight in high school, circa 1974; he also introduced me to Lord of the Rings, so thanks Dwight!)
Catan (more recent, obviously, but we play 3-4 days a week, every week, for the past 4 years)
Empire Builder (fond memories, and we’ve played off-and-on for about 20 years; we’re currently in “off” mode but I’m sure we’ll pick it up again soon)
Canasta (Mom taught me when I was around 12, and I taught the Mrs.; Mom’s been gone 29 years and now I’m crying)
I keep going back to Super Mario Bros. 3. I play through the entire game several times a year. I’m really good at it; it’s comfort food to me. I’ve played it regularly since it first came out, although I haven’t always had a system on which to play it.
I suspect I am the world’s best “Dino Eggs” player on the Commodore 64. Been at it since I was a kid, still play frequently to relax.
Looked around on some retro gaming boards once. Of the few people who play it (and discuss it on the internet), my high scores were orders of magnitude higher. Too bad there’s no money in it for me.