crunchyOwl: The Planet Crafter seems to have LAN/direct IP since the newest update.

Lifthrasil: Since it was quite depressing to create this thread about DRM on GOG, because I felt it necessary, I'll cheer myself up by making a list of games that still are 100% DRM-free on all parts, including multiplayer.
Yes, DRM-free multiplayer is possible! The 'but multiplayer requires online regristration of some form' excuse, that is often toted by apologists, is false. So here are the games that offer DRM-free multiplayer, either as LAN or Direct-IP (personal dedicated server). Note that many LAN games can also be played online via Hamachi or some other form of virtual LAN.

Since I hardly play any multiplayer, please help me out extending this list with good examples of games that don't require Galaxy or any online registration. They can allow online registration to some proprietary server (like Battlenet), as long as it is not required and the games offers DRM-free alternatives for multiplayer.

[In alphabetical order]

8-bit Armies (LAN)
Act of War (LAN)
Age of Civilization II (LAN only)
Age of Wonders (original): (LAN, hotseat and play-by-email)
Age of Wonders Shadow Magic (LAN, hotseat)
AI War 1 & 2 (direct IP)
American Conquest ( LAN )
Anno Series (Anno 1701 direct IP, 1602 direct IP on older machines, 1401 and, as far as I know, the others: LAN)
ARMA: Cold War Assault (LAN and Direct IP)
ARMA Gold (LAN and Direct IP, but with a unique serial key check - so not entirely DRM free)
Baldur's Gate 1&2 (LAN and direct IP for classic, direct IP for Enhanced)
Barony (direct IP and LAN Co-op)
Battle Isle: Incubation (LAN?, direct ip?, hotseat)
Blood Fresh Supply (direct IP)
Brothers in Arms Earned in Blood ( LAN )
Brothers in Arms: Road to Hill 30 ( LAN )
Brothers in Arms Hell's Highway (LAN)
Call of Juarez (LAN, internet via Doomseeker)
Call of Juarez Bound in Blood (LAN)
Civilization IV (LAN and direct IP)
Constructor Classic (IPX)
Crime Cities (LAN)
Crypt of the NecroDancer: SYNCHRONY (LAN, direct IP)
Dark Reign ( LAN )
Darkstone ( direct IP )
Deadlick II (TCP-IP)
Delta Force ( LAN )
Descent 1+2 (LAN and online)
Deus Ex (LAN and Direct IP)
Diablo+Hellfire (LAN)
Disciples II: Rise of Elves/Gallean Return (LAN and internet TCP/IP, hotseat)
Divinity Original Sin (LAN and direct IP)
Doom I+II (LAN)
Doom III (with a trick. Check the top rated review)
Dungeons and dragons Dragonshard (LAN)
Dusk
Empire Earth (LAN, direct IP with workaround)
Empire Earth II (LAN and Direct IP, but with a unique serial key check - so not entirely DRM free)
Evil Islands (LAN)
Fallen Haven: Liberation Day (LAN/WAN)
Factorio (direct IP, LAN, dedicated server)
Falcon 4.0 (LAN)
Fantasy Wars ( LAN )
Far Cry 2(LAN)
Flatout 1 & 2
Ghost Recon
Grim Dawn (LAN)
Hammerwatch (Direct IP)
Heretic: Shadow of the Serpent Riders (LAN)
HeXen: Beyond Heretic (LAN)
HeXen: Deathkings of the Dark Citadel (LAN)
HeXen II (LAN)
Icewind Dale 1&2 (LAN and direct IP for classic, direct IP for Enhanced)
Jagged Alliance: Deadly Games - (TCP-IP by using tunneling option from Dosbox on IPX feature in the game)
Jazz Jackrabbit 2: Secret Files (LAN, with JJ2+ also via internet, split screen)
M.A.X. (TCP-IP via IPX Dosbox, hotseat)
Master of Orion 2 (TCP-IP via IPX Dosbox, hotseat)
Medal of Honor Allied Assault and Pacific Assault (Direct IP and LAN)
Mobile Forces (LAN and Direct IP)
Neverwinter Nights Enhanced Edition - LAN or Internet Direct IP.
Old World ( Hotseat and LAN )
OpenTTD (direct IP)
Overload (LAN, direct IP)
Painkiller (LAN)
Powerslave (DOS) (TCP/IP LAN)
Project Zomboid (Direct IP)
Quake I + II
Quake III (LAN and direct IP)
Quake IV (LAN)
Rainbow Six ( LAN )
Red Faction (LAN and Direct IP)
Return to Castle Wolfenstein (LAN and Direct IP)
Risk of Rain(Direct IP)
Sacred 1 & 2 gold (LAN)
Saints Row 2 (LAN)
Saints Row Gat Out of Hell (LAN)
Saints Row The Third - The Full Package (LAN, only in the original, the remaster removed LAN and only left DRMed multiplayer in)
Scrapland (LAN, direct IP, dedicated Servers)
Screencheat (LAN, Hotseat)
Secrets of Grindea (direct IP can be toggled)
Serious Sam: First Encounter/Second Encounter
Sins of a Solar Empire(Direct IP)
Sir You are being Hunted (LAN)
Soldier of Fortune II (LAN and direct IP)
Space Empires IV (TCP-IP, PBEM, hot-seat)
Starbound
Stardew Valley (LAN)
Star Trek: Away Team (TCP/IP and LAN)
Star Trek: Starfleet Command III (TCP/IP and LAN)
Star Trek: Bridge Commander (TCP/IP and LAN)
Star Trek: Elite Force II (TCP/IP and LAN)
Star Trek: Voyager - Elite Force (TCP/IP and LAN)
Star Wars - Jedi Knight II (LAN and Direct IP)
Star Wars - Jedi Academy (LAN and Direct IP)
Star Wars - Republic Commando (LAN and Direct IP)
Streets of Rogue (Hotseat, LAN, even though the product page says that Galaxy is required)
Sudden Strike ( LAN \ IP )
Sudden Strike 2 (LAN \ IP )
Sudden Strike 4 (LAN )
SWAT 3 ( LAN )
Swat 4 (LAN, direct IP)
Terraria (dedicated server)
Titan Quest (LAN)
The Wheel of Time (LAN and Direct IP, although it requires dedicated servers)
The Planet Crafter ( LAN/direct IP)
Theme Hospital (IPX)
Torchlight 2 (LAN)
Trine 1+2+4 (direct IP and LAN (unsupported but possible), don't know about 3)
Unepic
Unreal (LAN, dedicated server, direct IP)
Unreal Tournament 99 (LAN and direct)
Unreal Tournament 2004 (LAN and direct)
Van Helsing final cut (LAN)
War for the Overworld (LAN and direct IP)
Warlord Battlecry
Warcraft II (LAN)
Warhammer 40k Gladius: Relics of War (Direct IP, hotseat)
Wasteland 3 (direct IP)
Wildermyth (direct IP)
World in Conflict (LAN and Direct IP)
Worms Armageddon (LAN, direct IP)
XIII classic with the 1.4.2.1 patch (dedicated server)


... to be continued ...

Same PC multiplayer only: (Hotseat, Split-Screen, Shared Screen etc. But no online or LAN)
Antihero (head-to-head hotseat)
Axis & Allies 1942 Online ( Hotseat )
Battle Chef Brigade (split screen)
Bonkies
Broforce
Cat Quest 2
Caveblazers
Cook, Serve, Delicious! 2!! (2-player split-screen co-op)
Crawl
Cuphead
Death Road to Canada
Emperor of the Fading Suns
Epic Pinball Collection
Fist Puncher
Fort Triumph
Hellpoint (Split screen)
Hot brass
Inkulinati (Hotseat)
JYDGE
Judge Dredd - Dredd vs Death ( Split -Screen )
Lovers in a Dangerous Spacetime
Missionforce: Cyberstorm
Moon Hunters
Neon Chrome (shared screen)
Oh...Sir!! The Insult Simulator
Outward
Pixeljunk Shooter
Police Stories
Relic Hunters Zero: Remix
Shadow Empire
Songs of Conquest (Hot seat)
Spelunky
Talisman: Digital Edition
Telepath Tactics
THE KING OF FIGHTERS 2002
Tonight We Riot
Tooth and Tail (Split screen)
Towerfall: Ascension
Trash Sailors
Tyrian 2000
Wizard of Legend
Worms Forts: Under Siege
X-Morph: Defense Complete Edition

I was notified, that there is already a list of older titles with single computer multiplayer.

For a list of games where DRM-free multiplayer can be added via mods or external tools, look here.

thiagott: Please add Wolfenstein: Enemy Territory to the list.
https://www.gog.com/game/wolfenstein_enemy_territory

We have been playing for a while
https://www.gog.com/en/news/for_todays_winter_sale_calendar_entry_join_our_dedicated_wolfenstein_enemy_territory_servers

Anyone can host using the game itself or using its dedicated server executable.
Anyone can join via direct IP, LAN, or using its server browser.
Connects to versions available on other platforms; Completely free.

The only small issue we found is that it doesn't seem to stop cheaters, so be careful while playing on public servers.

thiagott: Update to my previous message.
Today I tested OpenTTD. The GOG version seems to be updated to the latest stable version accordingly.

GOG version is currently 14.1.
The latest stable version from their official website is currently 14.1 as well.
F-droid version (Free & Opensource Android apps) is also updated to 14.1.

In my tests, they connect flawlessly between any OpenTTD that shares the same version number. Even if it's on Mobile, between different platforms (e.g. Windows+Linux+Android, arguably MacOSX too but never tested).
I managed to get Linux, Windows and Android versions to connect to each other, mixing GOG and non-GOG versions without any issue.

You can also download any version 14+ from their website and add Galaxy integration if you wish to use another version! (also available for Discord and Steam)

Please add Dedicated server and LAN to OpenTTD (besides direct-ip, also supported).
On Windows, Linux and MacOSX, append the -D parameter when executing "openttd".
Android version can also host, but it seems to lack the dedicated mode.

lanattic: Faces of war (LAN/directIP)

ConsulCaesar: User goggonzola2 has reported on the game specific subforum that Civilization III has working Direct IP multiplayer: https://www.gog.com/forum/sid_meiers_civilization_series/cant_load_lan_game_in_civ3_complete/post2