Formal:
* We would be delighted if you would join us.
* Please feel free to join us.
* We invite you to join us.
Informal:
* Come on in!
* Join the fun!
* Hang out with us!
* Let's get together!
* We're having a blast, come join us!
Enthusiastic:
* Don't miss out! Come join us!
* We're having a great time, you should come too!
* We're so excited to have you join us!
Specific to the situation:
* We're playing a game, want to play?
* We're having dinner, do you want to join us?
* We're going to the beach, are you coming?
The best way to say "come join us" will depend on the context and your relationship with the person you are inviting.