How To Use Telegram Without a Phone Number
3 Easy Ways to Join a WhatsApp Group Without an Invitation
2 Proven Methods: How to Use WhatsApp Without Phone Number 
5 Anonymous Texting Apps That Don't Require Your Phone Number
How to create and invite into a group
Let the SMS verification fail and choose to get a call instead.
A couple of the most popular messenger apps is Kik messenger as well as WhatsApp.
In theory, a sufficiently motivated government or law enforcement agency will have the resources to defeat your attempt at ultimate anonymity.