Black-box complexity theory provides lower bounds for the runtime of black-box optimizers like evolutionary algorithms and serves as an inspiration for the design of new genetic algorithms. Several black-box models covering different classes of algorithms exist, each highlighting a different aspect of the algorithms under considerations. In this work we add to the existing black-box notions a new \emph{elitist black-box model}, in which algorithms are required to base all decisions solely on (a fixed number of) the best search points sampled so far. Our model combines features of the ranking-based and the memory-restricted black-box models with elitist selection. We provide several examples for which the elitist black-box complexity is exponentially larger than that the respective complexities in all previous black-box models, thus showing that the elitist black-box complexity can be much closer to the runtime of typical evolutionary algorithms. We also introduce the concept of $p$-Monte Carlo black-box complexity, which measures the time it takes to optimize a problem with failure probability at most $p$. Even for small~$p$, the $p$-Monte Carlo black-box complexity of a function class $\mathcal F$ can be smaller by an exponential factor than its typically regarded Las Vegas complexity (which measures the \emph{expected} time it takes to optimize $\mathcal F$).