Is it possible to create a playable character who's a dragon in D&D 5e? Not a dragonborn, but an actual dragon.
I'm new to D&D and planning on playing D&D 5e because it's the newest.
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If your character has access to someone who can cast True Polymorph, the caster can transform your character into a dragon with a CR equal to or less than your level.
If you're casting it yourself as a 17th level Wizard or Bard, that means you have all the Adult Dragons avaliable. If someone else is casting it on you and you're a lower level, Wyrmlings are CR 3-4; Young are 6-9, Adult are 13-17.
Note that this will remove your class and racial features. If you're high leve enough to be able to become an Adult dragon this is normally a poor plan given you can throw around Wish/Miracle/etc and dragons don't (normally) have spellcasting.
If you're happy being simply dragon-like not an actual dragon, there are several options:
Dragonborn (phb p 32) "look very much like dragons standing in humanoid form, though they lack wings or a tail", have a breath weapon attack that they can use, and resistance to an element depending on what type of Dragonborn they are.
If you have Xanathars Guide to Everything, there are some Dragonborn-only Feats that make them more draconic, e.g. become tougher or improve their breath weapon (XgTE p74)
In the Feats for Races Unearthed Arcana (playtest material) there is also a feat that gives a dragonborn wings and a fly speed, but it is playtest material that did not make it into XGtE, so definitely ask your DM before you get set on having it.
The Draconic Bloodline Sorcerer makes you better at talking to dragons, better at doing elemental things of your bloodline, gives you a thin layer of scales, and eventually gives you the ability to manifest wings.