hands down without hesitation at the top of the list is, and i think will always be, the original 1997 Ultima Online!
it’s a massively multiplayer online role-playing game, based on medieval times, with so much freedom that your in-game actions could emotionally make or break other players (including me) in real life. i spent most of my high school time and some years after graduation playing this game. the grind was huge and your options were very extensive, but the actions during and following all that build-up could be EXTREMELY INTENSE at any unexpected time… safety? no such thing in UO. trust no one, except your friends, but even they - including the irl ones - might be greedy for UO at the expense of your friendship at times. no other game has given me such thrill as UO. i was super duper addicted, living a completely different life in that virtual-but-feels-so-real world.
next up has to be the original Ace of Spades:
is an explanation really necessary? ;D
third, probably the original Quake, especially the Threewave CTF mod:
because it was my first online multiplayer experience (other than a direct connect dial-up session with a friend in Doom II), and the first PC game i got super addicted to. i can still vividly remember the first time i noticed the irl outside sky turning orange as the morning sun was rising while i playing Quake Threewave CTF… i thought it was still just late, maybe midnight, so i was shocked to see daylight already.
4th: hmm, i think it’s a tie between GravBall/Infantry and Age of Empires II because although the gameplay of each was uniquely superb, both were rebound games for me the first time i broke up with Ultima Online.
of course, i ended up going back to UO later on because i missed it too much. :
but anyway, GravBall showcased and rewarded a player’s skill very well. it was one of many sub games in the Infantry package, and “Infantry” itself was a session-based RPG. very funnn times!
GravBall:
AoE2:

5th: the original Red Alert, simply because it was the first RTS i really got into:
and because… Tesla Coils. 
6th: the Sim City series, especially on SNES because that’s where i spent weeks upon weeks building my first city without cheats that i became very proud of and attached to, only to accidentally delete it one afternoon. when i realized what i did, instantly my heart sank and i felt depressed irl, the first game that affected me like that… i couldn’t even look at it for months.

7th: the Counter-Strike series. i’ve put countless hours into every version; original, Source, GO - all good. lots of fun and aggravating memories… online, at cafes, and at small personal LAN parties.

8th: the Battlefield series, every version from 1942 to the current newest, Battlefield 1. Battlefield: Bad Company 2 was my favorite, though, mainly because - not to brag or anything :
- i was unstoppable as a helo pilot on the Heavy Metal map. that game, and especially that map, is where i experienced the best skill i’ve ever had in any game.
i really enjoyed BFBC2 on other maps and modes too. between its “rush” mode and the Heavy Metal map, i think i had more intense moments in this game than any other first person shooter. nothing like your team suddenly sparking last-minute momentum to work together to turn the tables for the win!
and last but not least, i must mention Super Mario Bros. on NES! my first gaming experience other than arcade Pac-Man and Pong.

ok, ok, and Bubble Bobble on NES because it’s just good, simple fun and brings back lots of childhood memories, especially when i hear its music:







