Godzilla breathes concentrated streams of radiation. This appears to be a form of superheated vapor in the Showa era films, but becomes a pure energy beam in the 90s and beyond.
Omitted in Godzilla (1998) (where he technically has a "power breath" that simply makes things catch on fire rather than producing fire itself).
Godzilla has this power in Godzilla (2014), true to the original incarnation of him. Unlike the laser appearance from Toho's Heisei and Millennium eras, this incarnation's atomic breath is similar to the superheated vapour appearance from the Showa era. He uses it as the coup de grāce against the female Muto three times, but it's only when he force-feeds it to her that it has any real effect.
Ghidorah fires lightning bolts from its three mouths; these are sometimes called "gravity beams", and in Godzilla Final Wars they actually have an anti-gravity effect.
Destoroyah fires "micro-oxygen" rays that disintegrate matter.
Gamera breathes fire - a continuous stream in the original series, and exploding plasma balls in the modern films directed by Shusuke Kaneko. The sonic rays fired by frequent nemesis Gyaos straddle the line between Breath Weapon and Make Me Wanna Shout.
Really, this trope is so pervasive in the daikaiju genre that there is a great number of examples not listed here; attempting to include every one would likely prove fruitless.
In keeping with its roots in the above genre, Pacific Rim includes one Category 4 kaiju that spews blue corrosive slime from its mouth.
Another Batman example: The Red Triangle Circus Gang in Batman Returns has a performer dressed as a devil who contributes to the atmosphere of street terrorism by "blowing fire" on a display of jumbo stuffed animals in a toy store. Batman dispatches him by turning the Batmobile around so that the car's exhaust pipe is facing him and shooting out a jet of flame, setting the performer on fire.