The psychedelic "SAVE THE EARTH!" from Godzilla vs. Hedorah.
Reijiro Koroku's soundtrack to The Return Of Godzilla (or Godzilla 1985) is dark, gothic and brooding; Akira Ifukube meets John Williams at an Edgar Allan Poe convention.
The music for Godzilla vs. Biollante toned down the grim and added a bit of poetry to the tune.
"Bio Wars." Snippets of it played during BioMajor's escape scene leading up to SSS9 taking them all out, and a couple evacuation scenes. It's kind of a Super Sentai theme, with insane lead guitar, and snippets of the classic Godzilla theme to boot.
The Love Theme, only part of which was heard near the end, and Countdown. Much of the music was composed as suites which were then broken down into various pieces that are heard in the movie.
Watching Burning Godzilla rampage through Hong Kong to the chilling "The Godzilla March" remix called Hong Kong Destruction will send shivers through you.
Kow/Koh Otani's score for Godzilla, Mothra, King Ghidorah: Giant Monsters All Out Attack is also worth mentioning. A combination of classical and techno sound used to bring the past to the present, giving each monster its own new theme for the film.
For the Heisei era, it really can't get any more epic than Godzilla's Requiem. Epic, and really gives the death of Godzilla the gravity fans pretty much demanded as soon as the posters and trailers hit with the WHAM Line "On December 24th, Godzilla DIES".
Part of what makes Godzilla's awakening in Final Wars so epic is the music. As it builds up to his appearance, it goes silent as we see his face. He roars, and then the music just explodes with epicness!
Other Toho Films
As it has been mentioned before, Akira Ifukube's score for Varan and King Kong vs. Godzilla would be reused and modified for dozens of other films.
Though it was never really featured in any Godzilla media, the Blue Oyster Cult song can also qualify for this trope.
Many of the Unleashed themes are awesome to listen to, with the most awesome themes going to the GDF and Earth Defenders, but all the factions have awesome themes in there. Kiryu/Mechagodzilla 2's theme in particular, even if it's shared between two characters, is pretty cool-sounding, and Jet Jaguar has a theme that makes him feel badass. The crowning theme has to be Mecha-King Ghidorah's theme, though; it became the main theme of the game for a reason.