Let's take your Dragons example and see how the base classes fit with them (IMO):
Barbarian: The idea of a "savage" dragon is pretty good to me, but simply giving a dragon barbarian levels isn't enough, and the barbarian abilities do not necessarily fit well with the general abilities of a dragon. Rage would need to be pumped up to matter. Uncanny dodge sounds silly for a giant creature, and so does Trap sense. Damage reduction could also be pumped up. Indompitable Will is still good, as are the other Rage-related abilities.
Bard: The idea of a bardic dragon is really cool, IMO. The bard class could pretty much stay as it is.
Cleric: A Dragon could be devoted to one of the dragon gods, or to a particular philosophical concept. Turn/Rebuke undead could be granted to Dragons that actually worship something that has to do with life/death/undeath. Dragons should get domains.
Druid: A Druidic dragon is an interesting idea, but some abilities wouldn't really be appropriate: Animal companion, Woodland stride, Trackless step, etc. A Wild-shaped dragon is an interesting idea as well.
Fighter: The feat list would have to be changed a bit
Monk: Doesn't really apply, now, does it?
Paladin: Again, i'd have some difficulty accepting a Dragon paladin.
Ranger: doesn't fit.
Rogue: doesn't fit.
Sorcerer: the most obvious choice for a Dragon class.
Wizard: Also a good choice, but familiar rules would need to be revised.