I think the use case for the concept of Rust lifetimes which I can't do without is the tracking a compile-time mutable references and borrowing from them. Not for memory management, but because being able to guarantee that there is no other copy of the reference to the mutable object (thus borrowed mutability would be distinct type from owned/unique copy of mutability and of course borrowed immutability would be needed).
This will for example eliminate the requirement for immutability of the collection for those methods that modify the first-class union element type in my proposed complete solution to the Expression Problem.
It will also enable when to know whether closures form pure functions (for convenience and to eliminate unnecessary boilerplate) for my other thread about inversion-of-control of resource management.
Edit: follow-up.