The laws of thermodynamics can be extended to the nanoscale, where fluxes are fluctuating quantities. Little is known about extreme-value statistics of thermodynamic fluxes characterising the most extreme deviations from the average behaviours. Using Martingale theory, we study statistics of the negative records of stochastic entropy production in nonequilibrium steady states, and derive universal inequalities for such distributions of records. Furthermore, we explore the implications of our results in two non-equilibrium nanoscopic systems: single-electron transistors and molecular motors. We report on the experimental measurement of stochastic entropy production and of records of negative entropy in a metallic double dot under a constant external DC bias. Experimental results on the double dot confirm our theory and reveal a novel bound for the maximal heat that a mesoscopic system can absorb from its environment. We also explore our results in active biological processes and find predictions for the maximal excursion of a molecular motor against the direction of an external force.