Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly $\kappa^+$ many normal measures on the least measurable cardinal $\kappa$. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $\lambda$ strong compactness or $\lambda$ supercompactness measures on $P_\kappa(\lambda)$ can be exactly $\lambda^+$ if $\lambda > \kappa$ is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.