Dragon Games
Golden Treasure: the Great Green

Was there ever any doubt this would be the first game on the list? Golden Treasure is not only my favorite dragon game, it's likely my favorite game ever made. The gameplay is a mixture of visual novel, choose-your-own-adventure, and RPG, where you spend each day of your life as a dragon exploring, hunting, battling in simple rock-paper-scissors combat, and making decisions that can save or ruin your run.
Golden Treasure's strongest aspect, without a doubt to me, is its writing. The game does an incredible job at putting you inside the mind of a dragon, and the story and prose throughout Golden Treasure are downright magical, focusing on topics like life and death, nature, compassion, self identity, and much more. Golden Treasure still to this day has some of the most emotionally moving and satisfying moments I've experienced in a game, and I can't thank it enough for that.
Of course there are some downsides here too, such as the game's unskippable text, which can make replaying Golden Treasure tedious. Combat is unbalanced, with some abilities like power strike being very strong. Meanwhile other abilities like the true song of dominion are useless. Grinding out animal masteries is tedious, too, and the hunting minigame is janky.
The art and music of Golden Treasure are fantastic. Almost every part of this game feels polished and well thought-out, and it's clear that Golden Treasure was a labor of love more than anything else. The devs of Golden Treasure, Dreaming Door Studios, are currently working on a sequel to the game called Hierophant, so expect to see that one on this list too when it comes out!
The first playthrough of Golden Treasure is about 8-12 hours long, and at least double that if you want to see every ending. Your first run will be the hardest, but it gets easier after that.
Six Legends

There are going to be lots of visual novels on this list, but hey, they're a popular genre of furry games. It helps when they turn out to be this good, too.
Six Legends puts a strong emphasis on its main cast of dragons, as is clear from the name and cover art. Each character is lovable and well-written, but I have a special fondness for Blythe and Foam. The overall story is about acceptance and finding your place of belonging in this world, and it handles these messages well. Occasionally the dialogue felt too cheesy to me, notably during the last hour or so, but it's not a major detractor for me.
I admit I was hesitant when I first saw this one, due to its low budget appearance. But I'm very happy to be wrong. While this is mostly a visual novel, Six Legends also includes some minigames to mix up the gameplay. Have fun doing the music one on the hardest difficulty - it isn't lying when it says hard!
A playthrough of Six Legends takes around 8 hours.
The Dragon's Last Flight

Despite being in alpha as of now, Dragon's Last Flight is still good enough to recommend. This one is a pure visual novel where you play as an elderly man named Marius who encounters one of the last remaining dragons in the world.
Dragon's Last Flight may be the 2nd best written dragon game I've played after Golden Treasure. It touches on topics like aging and death in a thoughtful and mature way, and from such a rare perspective in visual novels as well. The artwork is nice, and I quickly fell in love with the 2 main characters and the relationship they build as the story goes on. While this game does have a few issues to iron out before release, like a lack of variety in backgrounds, I can already tell the ending is going to be a massive punch in the gut, and I wouldn't have it any other way.
It's hard to give a time estimate on this one, but the current alpha build is at least a few hours long.
The Mystery of Caketropolis

Caketropolis is an RPG-maker game where you play as a young gray dragon named Rimentus and try to solve the mystery of who burned down a bakery. While this game has its goofy moments (how can you not with a name like Caketropolis?), it was also created with a great message at its core. The story focuses primarily on the tensions between humanity and dragonkind, and Caketropolis has several surprising revelations that keep it interesting as you explore the small game world and unravel its secrets. The true ending is unexpected and controversial; I like it, but some people will definitely not.
The visuals of Caketropolis are very standard RPG-maker fare, but the music is great. General gameplay consists of simple puzzles and exploration. These do their job, but it's really the story and characters that take center focus here. One of Caketropolis's most unique features is Rimentus's ability to examine, smell, lick, or boop any object or person he finds. Try these out at various points, and you can get some pretty funny results!
A playthrough of Caketropolis takes about 3-4 hours.
(Fun fact: I currently own and use a Fleur mousepad.)
Perihelion

Perihelion is another visual novel in alpha, similar to Dragon's Last Flight. Only a single route is available right now, but it already has some lovable characters and nice art. The biggest criticism I have is that the dragons in this game don't feel like actual dragons due to a lack of dragon descriptors and behaviors (ex: not using wings, claws, tails, furniture and appliances not being built for dragons, etc). I hope this can be improved on with future updates.
Lark is a wonderful character, and her story is both personal and heartbreaking. It touches on the difficulties one can face juggling family, illnesses, and various other responsibilities all at the same time. I love Lark's feathered design and natural colors, too.
Scales of Stardust

A very basic, but fun (and free) 2D-platformer. This one has some nice movement options inspired by games like Ori, and I find it to be highly replayable. It's got good pixel art, stock music that manages to fit the level themes pretty well, and lots of speedrun potential. Our blue dragon protagonist is adorable, too.
Playtime is a brief 15-20 minutes, and even less on replays.
DRAGON: A Game About a Dragon

This is without a doubt the stupidest dragon game I've played, and I mean that in a good way. Dragon Game (as I'll call it from now on) is a level-based 2D platformer. Jumping and attacking are both slow and awkward, projectiles can hit you from offscreen, and some platforming sections contain blind jumps. Overall you'll run into plenty of jank while playing.
Where Dragon Game succeeds, however, is in its humor. As you can see from the screenshot above, the artwork was drawn with crayons, and this whole game is filled with funny dialogue and visual gags.
I wouldn't recommend this one if you're looking for a game with fast and fluid combat or platforming. But if you want to play as a dragon who speaks Haitian Creole and smokes weed with wizards while on a quest to rescue his human girlfriend, then I do recommend Dragon Game for some dumb, good fun.
Playtime is about 2-3 hours.
Honorable Mentions:
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Angels With Scaly Wings - I have great memories of this game from back in 2019, but now, years later, my thoughts on it are more mixed. The characters are generally likable, but the story is a mess of plot holes, time travel shenanigans, and poor worldbuilding (like Perihelion, the dragons here don't feel like dragons). It's still worth a try for most fans of dragon visual novels.
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Dragon's Wake - Decent, basic, clunky platformer in the same vein as DRAGON: A Game About a Dragon.