  It is a common perception that a sharp projective measurement in one side of
the Bell experiment destroys the entanglement of the shared state, thereby
preventing the demonstration of sequential sharing of nonlocality. In contrast,
we introduce a local randomness-assisted projective measurement protocol,
enabling the sharing of nonlocality by an arbitrary number of sequential
observers (Bobs) with a single spatially separated party Alice. Subsequently, a
crucial feature of the interplay between the degrees of incompatibility of
observables of both parties is revealed, enabling the unbounded sharing of
nonlocality. Our findings, not only offer a new paradigm for understanding the
fundamental nature of incompatibility in demonstrating quantum nonlocality but
also pave a new path for various information processing tasks based on local
randomness-assisted projective measurement.
