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.
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"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.