For us it was and ever shall be Age of Empires. Wololo!
I owe some of my high school social life to the fact I could beat anyone there at aoe, so I made friends with some of the older kids by teaching them good builds.
Come join us at reddit.com/r/aoe2 , it's fun!
Also, right when Doom came out, I remember my dad setting it up on work computers after work, and he would play deathmatches against his coworkers Dave and Mike, and they'd let an 8-year-old me play! Such fun times.
Also, one of the early GTAs had networked multiplayer, and I remember doing a fair bit of that when I was younger.