We report on the first measurement of exclusive Xi- and Xi0 photoproduction. The Xi- states are produced in the reaction gamma p --> K+ K+ Xi-, and the Xi0 states in gamma p --> K+ K+ pi- Xi0. Identification is made by the unique mass measured as the missing mass of the K+ K+ (or K+ K+ pi-) system using the CLAS detector at the Thomas Jefferson National Accelerator Facility. A systematic study of the excited Xi spectrum improves our understanding of the N* and Delta* states, since the Xi* states are related to them by SU(3) flavor symmetry. At the highest energies available at Jefferson Lab, we begin to find evidence for known excited Xi- states in the photoproduction process, and possibly new states at 1770 and 1860 MeV, although we do not have enough statistics to draw a strong conclusion. A search for the Xi5--(1862) pentaquark state seen by NA49 is made using the process gamma p -> K+ K+ pi+ X, but the result is inconclusive for lack of statistics.