We develop an analytical framework to derive upper bounds to light-matter interactions in the optical near field, where applications ranging from spontaneous-emission amplification to greater-than-blackbody heat transfer show transformative potential. Our framework connects the classic complex-analytic properties of causal fields with newly developed energy-conservation principles, resulting in a new class of power-bandwidth limits. These limits demonstrate the possibility of orders-of-magnitude enhancement in near-field optical response with the right combination of material and geometry. At specific frequency and bandwidth combinations, the bounds can be closely approached by canonical plasmonic geometries, with the opportunity for new designs to emerge away from those frequency ranges. Embedded in the bounds is a material "figure of merit," which determines the maximum response of any material (metal, dielectric, bulk, 2D, etc.), for any frequency and bandwidth. Our bounds on local density of states represent maximal spontaneous-emission enhancements, our bounds on cross density of states limit electromagnetic-field correlations, and our bounds on radiative heat transfer (RHT) represent the first such analytical rule, revealing fundamental limits relative to the classical Stefan-Boltzmann law.