One of the most famously awesome songs of the series comes from the original: Ginza has to be one of the coolest songs in the series.
The Battle Theme, which is pretty hard rocking for being one of the earlier games.
The Boss Theme, note that this link is for the track from the official soundtrack, and the full version isn't heard in game until it's reused in Shin Megami Tensei If.... Before then, the first part of the song is just looped.
Mysterious and foreboding, the opening is the perfect foreshadowing of what is to come.
The music that occurs during the scenes before and during the Conception. It starts with a somber yet peaceful piano intro during the conversation with Yuko before steadily begins to build up moments before the Conception occurs. Once the apocalypse starts in full sway, a majestic and epic organ plays throughout the following destruction of the world and the forming of the Vortex World, building up to a magnificent crescendo before being followed by an epic rock piece mixed with the organs to show that shit just got real.
When you hear this song, you know that badassery is just around the corner.
The Normal Battle theme. The fact that it's used during the Bonus Boss fight in Digital Devil Saga over the half dozen others themes is proof that it is that badass.
The best part of the theme (which you don't get when listening to a single recording of it) is that it has multiple variations of the main riff that are chosen randomly during the actual game. It really helps to keep it sounding fresh.
When you look death in the face, you don't hear yourself laughing. You hear this.
Kiyoharu's theme, which plays in Blasted Shinjuku. As a subdued arrangement of the Shin Megami Tensei I Law theme, it reflects the catastrophic damage that the forces of Law inflicted upon Tokyo, forcing humans to corral up in the last bastion of safety.
Sure, the first Persona has its share of problems, like a translation resembling something a Reston Ebola victim might vomit up, 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."
Eternal Punishment's "EX Final Battle" is very fitting for a Bonus Boss fight with Philemon. Notable in that the original's orchestral/techno sounds and PSP remix'spure rock keep the same melody, tempo, and general awesomeness, but sound really different from each other.
Finally, there's the Meguro's unused remix of "Knights Of The Holy Spear" for the PSP, which despite the title is actually a combination of "Knights Of The Holy Spear," "EX Final Battle," and the "Final Boss" theme mixed together in that order. Awesomeness to the third power! Though it was originally unused...
Jun's theme also known as "Blooming Flower" plays during much of the last act of the game accompanying Jun piecing together his memories. It's a solemn and tragic song perfectly befitting of the character.
Mt. Katatsumuri is a mysterious piece that plays during the part of the game where things start getting heavy.
Once you get to the last month of the game, the themes for the locations around the city get changed. And the city theme, Memories of the City is the best one of them all. The more somber-sounding theme really fits the darker look of the city, as well as the sheer weight of what the characters are preparing to do.
Living with Determination often plays during sad moments and when someone reaches the resolution they need to evolve their persona. A remix plays after New Year's Eve in the dorm. It pumps the player up to prepare for the arrival of Nyx.
This fan remix is an orchestral version, and has FREAKING BONGO DRUMS in the second verse!
The Reincarnation album brought us what is possibly the most epic version of "I'll Face Myself (Battle)" yet. So epic it gets reused in the Animation and Golden's fights with Shadow Mitsuo and Marie, respectively.
Speaking of Shadow Mitsuo, the Golden actually gives a special 8-bit remix of the theme called Shin Mitsuo Tensei.
Golden presents us with I'll Face Myself-Battle (Another version) when you have to fight Maire and her true form Kusumi-no-Okami. It's a dual remix with two version of I'll Face Myself and very epic at that too.
"The Bonds of Everyone's Souls" also came from the anime and like the title implies, it's also another Aria remix. Replace the opera singer with a choir and add some organs and guitars and you get a hopeful, awing song fit for a Moment Of Awesome.
The other final boss theme from the first game: Hari-Hara. The opening light guitar up to 0:21 makes you feel really cool and calm...until the really strong guitar kicks in, and the part from 1:03 to 1:24.
The airport theme from the 2nd game. One-Way Stairs. It starts out sounding like a fairly generic techno track, but around 0:30 it kicks into high gear.
Overclocked gives us Confusion as the map theme for the final fight of the 8th Days, with High touch being the new fast-paced final boss theme.
Cool Jam, the demon auction music, which also plays when Black Frost joins the party and functions as a sort of theme tune for him.
Devil Survivor 2
Mugen no Sekai, which can be translated variously as "Infinite World" or "Dream World". The opening theme of the game has a sweet and hopeful vibe to its deadly-sounding title.
The final battlefield theme: Akashic Records which is a rearrangement of the title theme.
The Septentriones' themes, both the obviously-titled field theme Septentrion and the battle theme Break Out. The field theme is appropriate for when you're readying yourself to attack a incomprehensible enemy, but when you get to the actual battle theme, you realise just how much weight the battle holds.
The remix of Septentrion used in the second phase of the final battle, Will of the Species, is truly excellent as well.
Speaking of the final battle, here is The World's Administrator, playing during battles against the last form of Polaris; giving the feel that he is no longer tolerant of your party of "foolish humans". All the sweeter when you kick his ass!
Not to mention Desperate Situation. You're fighting against unfathomable odds, but you have awesome music to pull you through.
A Gloomy Person. It's enough of a reason to talk to the Anguished One at every opportunity, even discounting the fact that his path is a "good" ending.
No matter what ending you take, it feels good and fulfilling when you get this theme here to greet you.