I have 4 kids in my game group but if one gets sick, I always look for games that play best with 3 people. A lot of the following games can be played with four people but you get the best experience playing with three.
Magic Labyrinth (my kids are slowly growing out of this one but it is an awesome game)
Near and Far (2-4, best 3 players, 60-120 min)
Above and Below (2-4 players, best with 3, 90 min)
Photosynthesis (2-4 players, best with 3, 30 – 60 min)
Welcome to the Dungeon (2-4 players, best with 3, 30 min)
Splendor (2-4 players, best with 3, 30 min)
Biblios (2-4 players, best with 3, 30 min)
Karuba (2-4 players, best with 4, 30 -40 min)
Dominion (30 min, best with 3 players)
San Juan (2-4 players, best with 3, 45 – 60 min)
Lords of WaterDeep (2-5 players, best with 3 or 4, 60 – 120 min)
Samurai (30-60 min)
Love Letter (2-4 players, 20 min)
Lanterns Festival (2-4 players, 30 min)
Kodama (2-5 players, best with 3 players, 40 min)
Tales Of The Arabian Nights (2-4 players, best with 3 players, 120 min)
Elfenroads (3-5 players, best with 4, 45 – 90 min)
Johari (2-4 players, 60-90 min)
Kanagawa (2-4 players, best with 4, 45 min)
Peloponnes Card Game (2-5 players, 25-40 min)
El Dorado (2-4 players, 30-60 min)
Port Royal (2-5, best 3-4 players, 20-50 min)
Century Spice Road (2-5, best 3-4 players, 30-45 min)
Century Golem (2-5, best 3-4 players, 30-45 min)
Dragonwood (2- 4 players, best with 3, 20 min)
Hey That’s My Fish (20 min)
Beasty Bar (20-30 min)
Tokaido (2-5 players, 45 min)
Small World (3-5 players, best with 4, 40-80 min)
Citadels (some people recommend it for 3 players, we have not tried it yet)–new, cheaper version here.
Istanbul (40-60 min)
Gubs (20-30 min)
Oh My Goods (2-4 players, best with 2, 30 min) with its extension Oh My Goods: Longsdale in Revolt
Five Tribes (2-4 players, 40 – 80 min)
Legends of Andor (1-4, best with 4 players, 60-90 min)
Rat-A-Tat-Cat (2-6 players, best with 4, 10-15 min)
Jamaica (3-6 players, best with 6, 30-60 min)
Ticket to Ride (30-60 min)
Catan (3-4 players, best with 4, 60-120 min)
Carcassonne (2-5 players, best with 2, 30-45 min)
Carcassonne: Hunters And Gatherers
Zooloretto (3-5 players, best with 4, 45 min)
Lost Cities (30-60 min)
Chateau Roquefort or under the title we own it: Burg Appenzell (2-4 players, 30 min)
Rummikub (60 min)
Machi Koro (2-4, best with 4 players, 30 min)
Istanbul (2-5 players, best with 4, 40-60 min)
Imhotep (2-4 players, best with 4, 40 min)
Hanabi (2-5 players, 30 min, co-op)
Sushi Go! (2-5 players, best with 4, 15 min)
Guillotine (2-5 players, best with 4, 30 min)
Five Crowns (3-7 players, best with 4, 30 min)
The Wizard (3-6 players, best 4-5, 45 min)
Azul (2-4, best with 2 players, 30-45 min)
Stone Age (2-4 players, best with 4, 60-90 min)
Survive (2-4 players, best with 4, 45-60 min)–although I have had a problem with 2 girls ganging up on one so it was not ideal.
Battle Sheep (2-4 players, best with 4, 15 min)
Terra Mystica (2-5, best with 4 bcs of more interaction, 30 min per player, so about 90 min for 3 players)
Scythe (1-5 players, best with 4, 90-120 min)
Castles of Mad King Ludwig (2-4 players, best with 4, 90 min)
New York 1901 (2-4 players, best with 4, 30-60 min)
Near and Far (2-4, best 3 players, 60-120 min)
Above and Below (2-4 players, best with 3, 90 min)
Cascadia (1-4 players, best with 2-3, 30-45 min)
Cartographers (1 to 9 players, best with 3-4, 30-45 min)
Ohanami (1-4 players, best with 3, 20 min)
Mystic Market (2-4 players, best with 3-4, 30-45 min)
Photosynthesis (2-4 players, best with 3, 30 -60 min)
Samurai Spirit (1-7 players, 30 minutes)
Eight-minute Empire Legends (1-4 players, best with 4, 18 – 32 min)
Mint Works (1 – 4 players, no difference, plays well with all numbers, 10-20 min)
Timeline Diversity (2-8 players, best with 4 players, 15 min)
Werewords (4-10 players, best with 6 – 7 players, 10 min)
Latice (2 – 4 players, 10-30 min)
On My Wish list:
Alhambra (2-5 players, best with 3, 45-60 min) or you can get a big box with all the extensions on Amazon here. After watching a few videos, I would go for the big box with all expansions. The best expansion is #2 but it is not available separately anymore. You will get all expansions in the Big Box.
Takenoko (2-4 players, best with 3, 45 min)
Santa Maria (1-4 players, best with 3, 45-90 min)
Pulsar 2849 (2-4 players, best with 4, 60-90 min)
Roll for the Galaxy (2-5, best 3-4 players, 45 min)
Age 14 and up
Mage Knight (1-3 players, best 1-2, 150 min)