unlimited conversations and unlimited messages
something works for an unlimited amount of data, in this case, infinite messages
These two statements may mean different things.
There is no such thing as an infinite number of entities in a computer system because computers have physical limitations, in terms of memory, storage, processing.
What you can do is to create testing charters detailing how you can explore these limitations.
Explore the application
with the phone XYZ
To discover how it handles keeping an extreme number of message loaded
Explore the backend service
to discover how it behaves when an extreme number of messages are stored for a single user
Then during your exploration, you will use your skills to try to force the target to these extreme conditions. Later you debrief the people who need this information so they can make decisions.
Test cases are good documentation for situations where you know what you want to know. You have a specific routine and a known expected result.
This is called checking. Testing is an open-ended performance to discover facts we don't know. Testing is exploration. (More about Testing versus Checking here)