Let's start with browser. Opera, made in Norway.
Search Engine. Ecosia, Qwant.
Social network: Mastodon.
Operating system: are you kidding right? Linux, Linus Torvalds is from Finland.
E-commerce company, just a few : Otto Group (bigger than uber, close in size to Baidu), Zalando, Booking.com, Digitec Galaxus (Swiss company, now selling in Switzerland, Austria, France, Italy; way better than Amazon).
Cloud computing platform: Hetzner, OVHCloud, SAP Cloud.
Financial services: Adyen (payment company, net income about 1/5th of paypal with 1/10th of the employees), Klarna, Revolut.
Entertainment company: if you accept video games as entertainment, Wooga from Berlin made Diamond Dash, Ninja Theory UK (Heavenly Sword, Devil May Cry), GameLoft is in Paris (Assassin's Creed mobile), Rovio (Angry Birds), Team17 UK (Worms, I know a bit dated but it's a classic), Crytek Germany (Crysis, Far Cry, Homefront), Supercell Finland (Clash of Clans), CCP Games Iceland (EVE Online, EVE Valkyrie), Arkhane Studios France (Dishonored, Bioshock 2), King Sweden (Candy Crush), RockStar North (Grand Theft Auto, Red Dead Redemption, Max Payne), CD Project Poland (The Witcher, The Witcher 3: Wild Hunt), DICE Sweden (Mirror's Edge, Battlefield), Hello Games UK (No Man's Sky).
Since we are talking about AI. DeepMind started and is still mainly based in UK, although it is now part of Alphabet.
DeepL, often better than Google translate.
ASML, the single company in the world that makes extreme ultraviolet (EUV) machines necessary to manufacture all the AI chips and latest processors.
ARM, based in Cambridge UK.
In the same vein: NXP Semiconductors, STM Microelectronics, Infineon Technology (originally Siemens semiconductor manufacturing division).
A few startups are doing good AI chips, e.g. Graphcore, GrAI Matter Labs.
IoT protocols: LoRA, LoRAWAN.
Open source: honorable mention to Redis, made by Salvatore Sanfilippo, fellow Italian like me.
OpenTitan: the root of trust chip that powers Google security chips in everything from Pixel phones to datacenter hardware, is a collaboration with ETH Switzerland, lowRISC Cambridge, G+D Mobile Security Germany, and other international companies (https://opentitan.org/).
A lot of industrial robot companies, which are essential to manufacture most stuff you buy on e-commerce sites, are from Europe: ABB (Switzerland) and KUKA (Germany) are the first and third companies worldwide by market for industrial robotics. Comau (Italy) is 5th. Stäubli (Switzerland) is 9th. Universal Robots (Denmark) is 10th. Notably, all the others in the top ten (Fanuc, Yaskawa, Epson, Kawasaki, Mitsubishi) are from Japan. None are from US.