I think my favorite boss music has to go to the Nemesis bioweapon from Resident Evil 3. He's a monster with one mandate: to seek out and destroy every last S.T.A.R.S. member in the doomed Raccoon City, and unfortunately for you the player, as the last surviving member in the city, you're hot on his radar.
The Nemesis, unlike every other bioweapon before him, feels seemingly indestructible. Your every encounter with him is on his terms. He's the one who stalks and engages you, not the other way round.
I'm gonna try and contextualize why I made these choices:
Nemesis Theme
The most common theme. You'll hear this early in the game. Imposing. The Nemesis is fully equipped with a Stinger Missile Launcher at this stage.
This is it. Up until this point you've been able to survive by running and hiding from the Nemesis. After destroying your rescue chopper and blocking your escape however, the only way forward is through him.
After a long hiatus and being ostensibly dead. The Nemesis comes fucking crashing through the church wall. His Stinger Launcher destroyed, body suit burned to shreds, and open wounds covering his body. He has lost none of his agility however and now has tentacles sprouting from his right arm and back. In this form Nemesis is more aggressive and the accompanying music represents that really well. Nemesis is stalking you again.
It's at this point that the tables turn. Having chased you into an Umbrella owned Disposal Facility, the Nemesis traps you in a room ready for decontamination. With only minutes to spare, and a seemingly impossible scenario, the Nemesis accidentally cracks open a valve and spills organic decomposition chemicals all over his body. Unable to regenerate from his wounds, the Nemesis can only drag his feet to chase you. This fight is hard, but you finally have Nemesis at a disadvantage. Using the valves to your advantage you take him down piece by piece. With no head or limbs left, he struggles one last time before collapsing and falling into a vat of chemicals.
After swimming in organic compounds, the Nemesis' disfigured carcass spills out in front of you. The monster goes through a massive twisted mutation and resumes it's pursuit. This fight is a literal fight against the clock. With nukes coming to "sterilize" Raccoon City, you're gonna have to take out Nemesis for good. Powering up an experimental Rail Cannon finally takes Nemesis out for good after three heavy shots tears him apart. After one final attempt, Jill grabs a Magnum and blows the thing apart.
Feel The Tension
Technically not boss music. This music plays when the Nemesis is nearby. It's nerve wrenching early in the game when you're so low on resources, never knowing if the monster is going to suddenly start running after you like a freight train.
Glad to see some of my picks have already been posted. I'm just gonna share a few lesser known tracks (or at least those which don't tend to crop up too often in these threads)