Quick thought: A game where the music gets more out of tune and broken as you tumble towards defeat.
Meteos did this, sort of. It would shift between different music clips depending on the scenario. If your blocks were almost to the top of the screen, the music would become frantic. It was all pre-recorded clips, being mixed in realtime. It wasn't generating music on the fly.
Super Mario Bros. did something almost sortof similar; when time was running out, the song would speed up to double-time, and at a higher frequency as a result, to encourage you to go faster.
But I'm thinking of… well, more like Eternal Darkness' insanity meter. I don't know if they did anything with the music in that game (I only played it briefly), but it would be fitting if, as your character started to go insane, notes in the music would start to shift off-pitch a bit, or it would occassionally hit the wrong note, etc.
This would be definitely be easiest with MIDI or MOD style music, rather than with MP3 or OGG where all the waveforms are mixed together and hard/impossible to un-mix.