Exceptional molecules and materials with one or more extraordinary properties are both technologically valuable and fundamentally interesting, because they often involve new physical phenomena or new compositions that defy expectations. Historically, exceptionality has been achieved through serendipity, but recently, machine learning (ML) and automated experimentation have been widely proposed to accelerate target identification and synthesis planning. In this Perspective, we argue that the data-driven methods commonly used today are well-suited for optimization but not for the realization of new exceptional materials or molecules. Finding such outliers should be possible using ML, but only by shifting away from using traditional ML approaches that tweak the composition, crystal structure, or reaction pathway. We highlight case studies of high-Tc oxide superconductors and superhard materials to demonstrate the challenges of ML-guided discovery and discuss the limitations of automation for this task. We then provide six recommendations for the development of ML methods capable of exceptional materials discovery: (i) Avoid the tyranny of the middle and focus on extrema; (ii) When data are limited, qualitative predictions that provide direction are more valuable than interpolative accuracy; (iii) Sample what can be made and how to make it and defer optimization; (iv) Create room (and look) for the unexpected while pursuing your goal; (v) Try to fill-in-the-blanks of input and output space; (vi) Do not confuse human understanding with model interpretability. We conclude with a description of how these recommendations can be integrated into automated discovery workflows, which should enable the discovery of exceptional molecules and materials.