Crowning Moment of Awesome: For whatever monster you completed adventure mode with. After taking on (presumably) every other monster in the world, you fight the alien leader, and beat the daylights outta him too!
Crowning Music of Awesome: While you often can't hear it due to the explosions and buildings collapsing, it is there. Most of it plays through the credits.
Capitol: Awesome music for a last stand against a stronger enemy.
Fanfic Fuel: Beating the game with Raptros or Zorgulon unlocks a Making Of Video, rather than an Origin Movie. Some fans have has great fun speculating their origins.
Goddamned Bats: The constantly-respawning Kineticlopses in Atomic Island, as well as the military's helicopters.
That One Boss: Goliath Prime, the first boss. The fact that he's big enough that other Kaiju consider him a boss should tell you something. The first third of the battle is a mild Guide Dang It moment (you have to catch the bombs he throws at you), and the rest of the fight alternates between him pummeling you, trampling you, or blasting you with cannons.
Also Cerebulon, the Final Boss. In his first form he fires a beam cannon that drains half your health, and about the only way you can avoid taking damage from it is to hide behind and object and let it take some of the hit. In his second form he invokes Instant Death Radius with a red lightning attack that drains half your health (to say nothing of how hard it is to actually damage this form). In his last form he uses a 12-hit combo that stunlocks you and, you guessed it, drains half your health.
That One Level: The Airfield. To summarize: a two-against-one battle. Small area with virtually no cover, but plenty of explosive objects to be thrown at you. Health pickups aren't more than 5 seconds away, no matter where you are, which means the enemy will eagerly grab them the second they start taking damage. And they think nothing of throwing explosives and spearing weapons non-stop.