The lore that's been built up is pretty complicated, and is important to the enjoyment of the games. However, without the context of release order, it can be confusing. So here's the canon order of the games (games not relevant to the ongoing lore aren't included).

• Charlie's Wake of Beginning

• Memory Game

• Egg One's Adventure

• Egg One's Adventure: Constantine's Revenge

• Egg Care Simulator

• The Vault

• GexCo Sports

• Egg One's Adventure: Beyond Cosmic

• Egg One's Adventure 64

• One Night at Constantine's

• Charlie's Wake of Beginning 2

• One Night at Constantine's 2

• One Night at Constantine's: BINARY

• Egg One's Adventure: AMBUSH (WIP)

• Egg Crossing

• One Night at Constantine's 3

• One Night at Constantine's: Secret of The Corruption

• One Night at Constantine's: Help Wanted (WIP)

A couple extra details to clear up any confusion.

• In the story, there was an old lore project made by GexCo, which used "real" egg AIs. In between the lore project and the gaming console, they were stored in a digital vault before escaping not much time into the gaming console being out.

• Between "The Vault" and "One Night at Constantine's 2", there was an event where the eggs were out in the gaming console, and had to be rescued by doing good enough in the games. All the games between those two (except EOABC and ONAC) have secret codes in them that can still be accessed.