Bound states in the continuum and exceptional points are unique singularities of non-Hermitian systems. In optical implementations, the former demonstrate strong enhancement of the electromagnetic field, while the latter exhibit high sensitivity to small perturbations. Hence, exceptional points are being actively investigated as next-generation optical sensors. However, at the nanoscale, their performance is strongly constrained by parasitic radiative losses. Here, we show that several bound states in the continuum can be merged into one exceptional point, forming a new kind of singularity. The resulting state inherits properties from both, namely, it does not radiate and shows extremely high sensitivity to perturbations, making it prospective for the realization of exceptional sensing at the nanoscale. We validate our theory with numerical simulations and demonstrate the formation of second- and third-order exceptional bound states in the continuum in stacked dielectric metasurfaces.