  Universality and scaling laws are hallmarks of equilibrium phase transitions
and critical phenomena. However, extending these concepts to non-equilibrium
systems is an outstanding challenge. Despite recent progress in the study of
dynamical phases, the universality classes and scaling laws for non-equilibrium
phenomena are far less understood than those in equilibrium. In this work,
using a trapped-ion quantum simulator with single-ion resolution, we
investigate the non-equilibrium nature of critical fluctuations following a
quantum quench to the critical point. We probe the scaling of spin fluctuations
after a series of quenches to the critical Hamiltonian of a long-range Ising
model. With systems of up to 50 spins, we show that the amplitude and timescale
of the post-quench fluctuations scale with system size with distinct universal
critical exponents. While a generic quench can lead to thermal critical
behaviour, we find that a second quench from one critical state to another
(i.e. a double quench) results in critical behaviour that does not have an
equilibrium counterpart. Our results demonstrate the ability of quantum
simulators to explore universal scaling beyond the equilibrium paradigm.
