The Spyro series is probably one of the most famous games that matches the sub-genre I've just outlined. Unfortunately, I could never really get into this franchise. Maybe it's the Saturday-morning cartoon vibe that turned me off to it. Regardless, I can confidently say cute purple dragons aren't my thing.
Moving from 3D to 2D, there's an old top-down, bullet-hell arcade shooter called Dragon Spirit that ate a lot my quarters growing up. It was kind of cool in that players could get power-ups that granted their dragon two or even three heads, thus doubling or tripling the firepower. However, I always felt like the dragon was way too fragile. Dragons are supposed to be well-nigh forces of nature yet in Dragon Spirit a few hits from anything is all it takes to crash and burn.
Another 2D game, this one from the side perspective, is Thanatos. It's an old Commodore 64 game that played like a mix of Choplifter and Wings of Fury, except instead of piloting a helicopter or Grumman F6F Hellcat you're a dragon. For better or worse, I never owned a C64 so my exposure to this game was pretty limited. Another exclusive to that home computer I did get to play over at a friends house was Dragon Riders of Pern. Based on a long running series of novels, this particular game was broken in to two distinct parts, a text-based strategy section and...well...let me explain the background for this setting a bit. Dragon Riders of Pern is actually sci-fi in that the "dragons" are genetically engineered by a bunch of space Amish. Every so often the orbit of Pern brings it near a bunch of gigantic filaments drifting in the void of space. This "thread" (as it's called) is extremely toxic to carbon based life and can fall to the ground with devastating effect. An entire country could be laid to waste unless the thread is destroyed as it descends through the atmosphere. How can it be destroyed? With fire! Hmmm...what flies and breaths fire? You guessed it! So getting back to the gameplay, it's basically a poor-man's version of Missile Command except the player is tasked with torching squiggly white lines with a dragon rather than intercepting ballistic nuclear warheads with an anti-missile system.
The PS3 near-launch title Lair attempted to do some experimental things with the Sixaxis controller that sounded good in theory, but didn't work in practice. Maneuvering dragons through tilt controls made the game essentially unplayable and even when the developers patched in analog controls the camera and story still stunk. It's a real shame because Lair had some nice looking dragons, as well as smooth transitions between air and ground combat.
There's all kinds of weird outliers I could mention. Take the Panzer Dragoon series for example. It has dragons but drops any semblance of fantasy in lieu of a futuristic atheistic. Another example is Drakan which might have been interesting if the visuals weren't so bland and generic. The list of be-the-dragon video games goes on, but there has never been one that really nails the feel of being the alpha predator of mythical beasts. That said, I would recommend one game without hesitation, the pseudo-arcade title TROGDOR! It's not good, but it is silly...and better still completely free. So even if you don't like it no loss...except maybe a few minutes of your time.