Well, GOG sells a lot of commercial games and doesn't require online connectivity for anything marked as "DRM free". Tend to be older. Once you download it, no link to the service required.
I think that all the -- be they free or commercial -- games on itch.io don't require signing up for a service, unless the game itself has some sort of service. I don't have specific recommendations there, though.
Games bundled in a Linux distro won't require a service.
There are open-source games.
I personally like Cataclysm: Dark Days Ahead, which is a very deep open-world roguelike set in a post-apocalyptic world with zombies. Steep learning curve, as a warning, but you can do all sorts of stuff. NPCs, build bases and set up electrical power, build ground vehicles, boats, and rotary-wing craft. Vehicles can carry other vehicles, can have video cameras, turrets, armor, various sorts of lights, rams and other melee weaponry. Bionics, mutations, skills, farming, crafting, quests, music and sound packs, graphical tiles. Martial arts. Contains probably more real-world firearms than any other game I know of, does stuff like multiple optics, various stock and handle modifications, powder fouling. Moddable melee weapons. Artifacts. Mods to add spells, psionics, and various magic items. Traps and static defenses. Cooking, brewing, drugs, alcohol, various types of clothing. Explosives. Waifu body pillows. Regional weather simulation. Heating and cooling. Lovecraftian stuff. Radiation. Remote-controlled vehicles. Senses including smell, hearing with temporary and permanent impairment modeling, infrared, vision to see magnetic fields, light-amplification optics, eye dilation simulation when entering different light levels. Vehicle-mounted battery chargers, kitchenettes, water tanks, rainwater collection systems, water purification systems. Radios. Various factions of enemies, some of which fight each other. Bandits. Lockpicking, teleportation, various types of diseases, parasitic and fungal infections, various types of poisoning. Hacking. Furniture. Various types of psychological conditions. Gasses, gills, skates, broken limbs, stances, folding bicycles, body part level encumbrance, container size maximums (including modeling things like mesh bags that can't contain small items and waterproof containers that protect things that are destroyed by immersion in liquid), pockets in clothing, various types of holsters and sheaths that can be worn on various places on the body. Pain, guilt, cannibalism, music. It's got a lot of stuff. There's a build on Steam now pre-set up with graphics and sound if you want to donate, but you can also just download the builds from the dev site for free. Has mobile builds, but I think that it really benefits from the computational power of a PC, as well as a keyboard.
Dwarf Fortress also has a steep learning curve, is a colony simulator. Not open source, but free, also deep, many hours you can spend there.
Shattered Pixel Dungeon is an open source roguelike, relatively shallow learning curve. Really aimed at touchscreen devices like smartphones, but has computer builds, has support for keys and stuff. See [email protected].
Mindustry is an open-source factory automation game in the vein of Factorio. Works on mobile or PC platforms.
I've only played Unciv on smartphone, but apparently it also has PC builds. It's an open-source reimplementation of Civilization V, sans all the pretty graphics and animation and music and such. One of the deeper games I think you can get on a phone.
Someone else mentioned Minecraft. I think that that requires an account with the service these days, though Luanti -- until recently known as Minetest -- is a similar, open-source project that does not.
Dungeon Crawl: Stone Soup is a tough roguelike known for being well-balanced, with the devs stripping out unnecessary stuff and streamlining it. I don't play it much these days, but I've enjoyed it in the past.
Endless Sky is an open-source clone of Escape Velocity for the classic Mac, a 2D space exploration, fighting, and trading game. I don't play it much, but I think that it's worth a look if you've never played it.
Battle for Wesnoth is an open-source tactical hex-grid game. Characters can level up and gain abilities. Can be played on mobile OSes, though I think that it really benefits from a mouse.
I am not personally all that into OpenTTD, an open-source game based on Transport Tycoon Deluxe, but I have played it and have seen many people who are super-into-it.
I've played and enjoyed the open-source 0 A.D., some time back, but last I played it, it had a bunch of work still to be done. An Age of Empires clone.
There are a handful of open-source RTS Total Annihilation-inspired games based on the open-source Spring engine, like Beyond All Reason.