Sure, the first Persona has its share of problems, like a translation resembling the product of a drunk interpreter, however, there is one thing that makes the game such a cult hit, the boss battle song Dead Line. It also got remixed in the Persona 2 Eternal Punishment remake, renamed as "The Brink of Death."
"School Days" from the PSP remake is exceptionally catchy.
"Let Butterflies Spread Until The Dawn," the overworld theme from the PSP remake, captures the ominous feel of traveling through a demon-infested version of your hometown pretty well.
"Child Abuse". Don't let its title fool you, it's anything but serious. But it is bloody catchy.