This is obviously not complete yet. It's also a little messy. Please understand.
Will list every game that supports achievements on this platform.
Will list any notable information about the game and comparative facts to their Steam counter-parts.
I'll link to achievement guides where applicable.
If you see any inaccurate or missing information then please let me know. Any feedback is welcome.
I know GOGMixes exist but I prefer forums by a mile. Maybe when this thread is more complete I'll make one.

If you wanted to see a clean list of games that support achievements:
[exophase link]
You can also make an account there to track your progress in games and see all kinds of other fun stats.

Then of coarse if you wanted to buy achievement games:
[GOG store link]

#
8-bit Armies
One less achievement than Steam version.

8-bit Hordes

8-bit Invaders

80 Days

A
A Boy and His Blob

Absolver

AER - Memories of Old

Agatha Christie - The ABC Murders

Ancestors Legacy

Anima: Gate of Memories

Aragami
Twelve less achievements than Steam version.

Ashes of the Singularity: Escalation

Ash of Gods: Redemption

B
Banner Saga

Banner Saga 2

Banner Saga 3
Twelve less achievements than Steam version.

BASEBALL STARS 2

Batman - The Telltale Series

Batman: The Enemy Within - The Telltale Series

Battle Brothers

Battlestar Galactica Deadlock

BATTLETECH

BattleZone 98 Redux

Battlezone: Combat Commander
Eighteen less achievements than Steam version.

Beholder

BLACKHOLE

Blackwood Crossing

Bloodrayne: Betrayal

Bombshell

Breach & Clear
Twelve less achievements than Steam version.

Breach & Clear: DEADline Rebirth

C
Caladrius Blaze

Candle

Carmageddon: Max Damage

Caveblazers

Celestian Tales: Old North

Children of Zodiarcs

Cook, Serve, Delicious!

Cook, Serve, Delicious! 2!!

Corpse Party

Cosmonautica

Craft The World

Crookz - The Big Heist

Crypt of the NecroDancer

Cultist Simulator

Cuphead

D
Darkwood
Dead Cells
Deadly Tower of Monsters, The
Dead Synchronicity: Tomorrow Comes Today
Deathtrap
One less achievement than Steam version.
Deponia
Deponia 2: Chaos on Deponia
Steam calls the game "Chaos on Deponia".
Deponia 3: Goodbye Deponia
Steam calls the game "Goodbye Deponia".
Deponia 4: Deponia Doomsday
Steam calls the game "Deponia Doomsday".
Detective Gallo

Deus Ex GOTY Edition
Achievements are exclusive to GOG version.

Devil Daggers
Achievement Guide:

Dex
Diluvion
Divinity: Original Sin - Enhanced Edition
Divinity: Original Sin 2 - Definitive Edition
Donut County
Doom 3: BFG Edition
Dragon's Dogma: Dark Arisen
Dungeons 2
Dungeons 3
Dying Light: The Following - Enhanced Edition
Steam calls the game "Dying Light".
This bundle is exclusive to GOG.

MachiaVillain

Offworld Trading Company
Three more achievements than Steam version.

Reservoir Dogs: Bloody Days
No longer sold on GOG or Steam.

Rezrog

Tahira: Echoes of the Astral Empire
Tales from the Borderlands
Tempest
Steam calls the game "Tempest: Pirate Action RPG".

The Adventure Pals
Seven less achievements than Steam version.

The Bureau: XCOM Declassified
The Escapists: The Walking Dead
The Incredible Adventures of Van Helsing - Complete Pack
Steam calls the game "The Incredible Adventures of Van Helsing".

The Incredible Adventures of Van Helsing: Final Cut
The Incredible Adventures of Van Helsing II - Complete Pack
Steam calls the game "The Incredible Adventures of Van Helsing II".

The Incredible Adventures of Van Helsing III
THE KING OF FIGHTERS '98 ULTIMATE MATCH FINAL EDITION
THE KING OF FIGHTERS 2002 UNLIMITED MATCH
THE LAST BLADE
The Last Door: Season 2 Collector's Edition
The Legend of Heroes: Trails in the Sky
The Legend of Heroes: Trails in the Sky SC
The Legend of Heroes: Trails in the Sky the 3rd
The Legend of Heroes: Trails of Cold Steel
The Legend of Heroes: Trails of Cold Steel II
The Lion's Song
The Long Reach
The Original Strife: Veteran Edition
The Real Texas
The Solus Project
The Walking Dead: A New Frontier
No longer sold on GOG.

The Walking Dead: Michonne - A Telltale Miniseries
No longer sold on GOG.

The Walking Dead: Season 1
No longer sold on GOG.

The Walking Dead: Season 2
No longer sold on GOG.

The Walking Dead: The Final Season
No longer sold on GOG.

The Whispered World: Special Edition
The Witcher 3: Wild Hunt
This is a stackable game with the same achievements as "The Witcher 3: Wild Hunt - Game of the Year Edition" but you must buy DLC for some achievements.

The Witcher 3: Wild Hunt - Game of the Year Edition
This version is no longer sold on Steam.
This is a stackable game with the same achievements as "The Witcher 3: Wild Hunt"

The Witcher Adventure Game
Thirteen more achievements than Steam version.

Think of The Children
This Is the Police
This Is the Police 2
This War of Mine: Soundtrack Edition
Steam calls the game "This War of Mine".

Tokyo Xanadu eX+
Tooth and Tail
Tower of Time
Transport Fever
Treasure Adventure World
One less achievement than Steam version.

Trine 3: The Artifacts of Power
Tropico 5
Turok
TWINKLE STAR SPRITES
Tyranny

Unforeseen Incidents

Vaporum
Victor Vran
Steam calls the game "Victor Vran ARPG".

Vikings - Wolves of Midgard

War for the Overworld
Way of the Samurai 3
West of Loathing
Where The Water Tastes Like Wine
Wizard of Legend
Wolf Among Us, The
Wonder Boy: The Dragon's Trap
World to the West
Worms World Party Remastered
One less achievement than Steam version.

Xanadu Next
XCOM: Enemy Unknown Complete Pack
Two less achievements than Steam version.
This bundle is exclusive to GOG.

X Rebirth

Yooka-Laylee
Ys: Memories of Celceta
Ys: The Oath in Felghana
Ys Origin
Ys SEVEN
Ys VI: The Ark of Napishtim

Ziggurat
Zombie Night Terror
Two less achievements than Steam version.

Zwei: The Arges Adventure
Zwei: The Ilvard Insurrection

Didn't realize this would auto-merge double-posts.

Newest games to have achievements added:
(Games already on the service can have GOG achievements added after the fact, so these aren't necessarily new games to the service.)
(ordered from newest to oldest, going as far back as September 2018.)

Sorcerer King
Hybrid Wars
BioShock 2 Remastered
BioShock Remastered
Fallen Enchantress: Legendary Heroes
Parkitect
Book of Demons
Monster Prom
Holy Potatoes! A Spy Story?!
Guns, Gore & Cannoli 2
Megaquarium
Candleman: The Complete Journey
Star Control: Origins
Asterix & Obelix XXL 2
Corpse Party: Book of Shadows
Redout: Solar Challenge Edition
Orphan
Solstice
Staxel
My Memory of Us
Project Hospital
Desert Child
Bad North
Mutant Football League: Dynasty Edition
My Brother Rabbit
The First Tree
ADOM (Ancient Domains Of Mystery)
Flashback
Egypt: Old Kingdom Demo
Depth of Extinction
Darksiders III
Cinders
Pathfinder: Kingmaker
Predynastic Egypt
Predynastic Egypt Demo
Egypt: Old Kingdom
Leisure Suit Larry - Wet Dreams Don't Dry
Lords of the Fallen Game of the Year Edition
Thronebreaker: The Witcher Tales
GWENT: The Witcher Card Game
Project Warlock
Guns, Gore & Cannoli
Urban Empire
A Hat in Time
THE KING OF FIGHTERS XIII GALAXY EDITION
Hellblade: Senua's Sacrifice
Ken Follett’s The Pillars of the Earth
Fallout: New Vegas Ultimate Edition
Ys I
Ys II
Doom 3: BFG Edition

Top ten achievement unlockers on GOG:
(to be in the running add your name to the ExoPhase site leaderboard)
Placement | Name | Achievements unlocked
1st misharapp 228
2nd shlomi83575 150
3rd jncreed2000 147
4th Tellus 147
5th BeatriceElysia 146
6th ThomasPierson 129
7th Tommy1984 125
8th Hauntshade 110
9th Kabouka 108
10th Kamui18 99

Shadow Warrior 2, Shadow Warrior Classic Redux, Outlast 2, Party Hard, Pharaonic, Event[0], and Gekido Kintaro's Revenge and many SNK games also have Galaxy achievements, while Celstian Tales: Old North no longer advertises achievements. The Galaxy page lists them for Celestian Tales, but they are unattainable.

Thank you for this information.