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Complex phenomena know several benchmarks, or ΄hard΄ and to date unsatisfactorily understood problems. Human heart rate control is such a complexity benchmark in biophysics, consistently defying full explanation. In our recent work, heart rate regulation by the autonomic nervous system has been shown to display remarkable fundamental properties of scale-invariance of extreme value statistics [1] in healthy heart rate fluctuations, ubiquitously observed in physical systems at criticality, and to undergo a phase transition as a result of altered neuro-regulatory balance [2]. Most recently, we have shown this behaviour to depart from the critical scale invariance in the case of life-threatening congestive heart failure (CHF). Our new index, derived from the non-Gaussianity characteristic, has proved to be the only meaningful one among all known heart rate variability (HRV) based mortality predictors [3]. We speculate on possible mechanisms for the increased variability and complexity of heart rate for life-threatening CHF, as reflected in the intermittent large deviations, forming non-Gaussian ΄fat΄ tails in the probability density function (PDF) of heart rate increments and breaking the critical scale invariance observed in healthy heart rate [1,2]. [ABSTRACT FROM AUTHOR]