  Quantum metrology shows that by exploiting nonclassical resources it is
possible to overcome the fundamental limit of precision found for classical
parameter-estimation protocols. The scaling of the quantum Fisher information
-- which provides an upper bound to the achievable precision -- with respect to
the protocol duration is then of primarily importance to assess its
performances. In classical protocols the quantum Fisher information scales
linearly with time, while typical quantum-enhanced strategies achieve a
quadratic (Heisenberg) or even higher-order polynomial scalings. Here we report
a protocol that is capable of surpassing the polynomial scaling, and yields an
exponential advantage. Such exponential advantage is achieved by approaching,
but without crossing, the critical point of a quantum phase transition of a
fully-connected model in the thermodynamic limit. The exponential advantage
stems from the breakdown of the adiabatic condition close to a critical point.
As we demonstrate, this exponential scaling is well captured by the new bound
derived in arXiv:2110.04144, which in turn allows us to obtain approximate
analytical expressions for the quantum Fisher information that agree with exact
numerical simulations. In addition, we discuss the limitations to the
exponential scaling when considering a finite-size system as well as its
robustness against decoherence effects. Hence, our findings unveil a novel
quantum metrological protocol whose precision scaling goes beyond the
paradigmatic Heisenberg limit with respect to the protocol duration.
