A look at FreedomBox 0.2

while a Dynamic DNS provider is the nice way to do things, if you just have the system connect out to something that can record and display it's address, you can work-around the lack of DNS (IP addressing still works :-)

This can be as crude as sending an e-mail to several outside accounts and looking at the header information These can be free accounts.

a bunch of different people could host websites that the home system periodically report it's IP to, and the user can then look at them.

DDNS is the slickest way to do it, but if you are concerned about the reliability, cost, or legal reliability of DDNS providers, setup your own fallback method.