You're a dragon and you start out in a brown underground cave. There's
a rainbow-flashing egg you can pick up. You flap your wings by tapping
the fire button on the joystick. The cave has two or three rooms and
has an opening into an overworld, which you can fly around in. On
certain screens there's a very fast moving pegasus? that kills you.
That's about as far as I got..

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Request for Question Clarification by politicalguru-ga on 20 May 2006 11:18 PDT

Could it be this one: 
Dragon Hawk
<http://en.wikipedia.org/wiki/DragonHawk> ?

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Clarification of Question by ostranenie-ga on 22 May 2006 20:46 PDT

No, that's a pretty cool game, but that's definitely not it. In the
game I'm talking about you start out in an underground cave and
there's a glowing egg you can pick up.. The cave has two or three
rooms and an exit above in one of the rooms. Once you go into this
exit you go into a sky-world with clouds, and I don't know much more
beyond that because I usually was killed by a pegasus-looking thing
and had to start over. The game is a bunch of static screens - you go
from screen to screen (up, down, left or right) instead of scrolling
as in Dragon Hawk. Thank you though!

Here are some possibilities:

Dragon Skulle
http://www.lemon64.com/index.php?mainurl=http%3A//www.lemon64.com/reviews/view.php%3Fid%3D45

Dragon's Den
http://www.lemon64.com/index.php?mainurl=http%3A//www.lemon64.com/reviews/view.php%3Fid%3D45

Rainbow Dragon
http://www.lemon64.com/index.php?mainurl=http%3A//www.lemon64.com/games/details.php%3FID%3D2077

Saint Dragon
http://www.lemon64.com/index.php?mainurl=http%3A//www.lemon64.com/reviews/view.php%3Fid%3D287

It's joust (but it's not a dragon, so the other comment is probably off subject).

Wikipedia has an in-depth article on it:

http://en.wikipedia.org/wiki/Joust_(arcade_game)

It's not Joust..... it's a game where you can go from screen to screen
by moving to the edge. Same principle where you tap the joystick
button to flap your wings and gain lift though.

http://www.lemon64.com/games/popup_screen.php?name=dragons_den_01.gif

reckon that's it