StarCraft and Counter-Strike 1.6 have been around forever and still have hundreds of thousands of people playing them online + major tournaments with serious money and huge amounts of professionals. Who knows how SC will handle its sequel but CS has actually rebuffed attempts by Valve to obsolete it with prettier but uninspiring sequels (CZ/Source).
Megawads are still coming out for Doom 2. There's still a thriving Doom 1-2 + custom speed run community. It's ugly as sin but there aren't any games like it anymore that combine ridiculous player movement and hundreds of interesting monsters -- plus the BFG is just way too fun.
I played Chrono Trigger several years ago for the first time. I'm not a big fan of RPGs in general but I really liked CT. Amazing music. Makes for even more amazing OC remixes.
Games which didn't age well?
Diablo 1. Solid game if only for the dark and foreboding atmosphere and some solid enemy combat designs but the graphics are so bad they hurt gameplay -- good luck trying to find items on the pixelated gray brown floor. And you can't run. Running like D2 would probably hurt the claustrophobic style, but still, it'll make you want to claw your eyes out in town. There could be some compromise here.
Ocarina of Time was somehow my first Zelda (followed by Majora) but I was playing Mario 1 when I was a little kid. I somehow missed the whole Zelda thing. Went back and played A Link to the Past and it's much better. OoT's mind numbingly slow text, hollow cinemas for the sake of having cinemas, and the idea of the empty and boring Hyrule Field being impressive date it badly. It's also ridiculously easy. It still has good qualities (dungeon design, some music) but it lost a lot of that 1998 luster.
Old school Resident Evils. Embarrassing to play anymore. Like parodies of paraodies which forgot the original source material. The coolness of zombies / science gone wrong + atmosphere is soon interrupted by obnoxious door loading sequences, dialogue so bad it's bad (it doesn't come all the way around the dial to funny, as some propose), the inventory, and strange controls.
Metroid 1 is like a pioneering player hating ROM hack attempt. It's not so bad if you know what to do, but that first, blind playthrough? Good luck. Let's have several seemingly indentical rooms, except one will have a random section of wall / floor be bombable which leads to the next area. Or falling through fake lava. Catchy, if repetitive, music. Limited aiming will drive you crazy.
GoldenEye / Perfect Dark are still fundamentally good games but playing them will give you a headache. It's like a power point presentation.
I'm not really sold on Half Life 1 still being as good as back in the day. It seems pretty bland nowadays I think. The enemy soldiers are really outdated, like watching an old '50s monster movie where you can still see the strings. Props for the last boss being a giant space fetus though.