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. 
The Pokemon Contests (using the Pokemon's moveset to appeal to judges, rather than combat) from Ruby Sapphire was my favourite part about those games. I wish they'd bring them back.
My partner LOVES Pokemon contests but not the bad ORAS remake ones. Their favorites were in Diamond and Pearl because you dress up your Pokemon and they thought it was cute. They say "there are dozens of us"
I skipped Diamond and Pearl! Gunna have to play them now they sound dope. And your partner has good taste
I miss battle revolution. The fully customizable trainers were great. I had a body builder I have the little kid lines for. "BE MY FRIEND" with the Pikachu facepaint on the Schwarzenegger body was a crowd pleaser