  Sequential weak measurements of non-commuting observables is not only
fundamentally interesting in quantum measurement but also shown potential in
various applications. The previous reported methods, however, can only realize
limited sequential weak measurements experimentally. In this Letter, we propose
the realization of sequential measurements of arbitrary observables and
experimentally demonstrate for the first time the measurement of sequential
weak values of three non-commuting Pauli observables by using genuine single
photons. The results presented here will not only improve our understanding of
quantum measurement, e.g. testing quantum contextuality, macroscopic realism,
and uncertainty relation, but also have many applications such as realizing
counterfactual computation, direct process tomography, direct measurement of
the density matrix and unbounded randomness certification.
