EDIT: Seems dynamic music is back in style in some very recent games, many of which I haven't really played yet. Good. 
For me, it's dynamic music, the kind that some games had that adjusted moment by moment to what was happening in the game.
The best-known example of this in the 90s game TIE Fighter, where the moment more enemy (or allied) ships showed up the music would have a little additional flourish to acknowledge the shift in battle. There were pre-battle tension tracks, battle music, complications of battle, grandiose flourishes for the arrival of enemy or even allied capital ships, and victory and failure music all ready to flow into the next seconds of the game.
A lesser-known but still excellent example of this was in Ultima Underworld and its sequel, where drawing a weapon had its own special "preparing for battle" tension music, getting attacked had a jump-out-of-your-skin joltingly sudden musical start that actually scared me as a kid when I got ambushed, music for battles going well, going poorly, victory and defeat.
I wish more games did those sort of second by second musical changes, but they've sort of fallen out of fashion for the most part. 
Virtual Pets in games. Like the mag system in Phantasy Star Online or the Chao in Sonic Adventure, such a fun way to waste time, especially since I already enjoy v-pets.
Allow me to shamelessly plug my own work
Cool comrade, bookmarking this
Damn, sick af. I have a pokemon simulator I was working on that didn't get past the initial stages, but I would love to eventually get it to a more "playable" state. If only I had the time and drive
Oh, the PSO mags. Takes me back.
Final Fantasy VIII even had a chocobo v-pet you could put on the pocket station but it never came to the states.
The PC CD-ROM version had it as a desktop game
Most recent one that comes to mind for me is in Monster Hunter World, where you have a sidekick cat that you can dress up. Still not a full pet experience like the Chao garden tho.