Mean age and the age spectrum, transport diagnostics independent of photochemistry, have been put to good use diagnosing stratospheric circulation and evaluating models. These diagnostics reveal how long air took to reach a stratospheric location from the troposphere, but they provide no direct information about where the air traveled en route. I define a complement to the age spectrum to explore how stratospheric transport timescales are related to path histories. Just as one constructs the age spectrum by distributing components of an air parcel according to transit time since last tropospheric contact, one can distribute the components according to the maximum height achieved en route to the parcel. Combining the age spectrum and the maximum path height distribution, one has a joint distribution of transit time and maximum path height. These ideas are illustrated with an idealized model of stratospheric transport, as well as a general circulation model. Transport features in the idealized model that strongly affect mean age affect the distribution of maximum path heights relatively weakly, especially in midlatitudes. The mass fraction of an air parcel at r that passed above some specified height z above r is strongly constrained by the mass continuity of stratospheric regions and is only more weakly affected by the circulation. This represents a limit on the relationship between mean age and trace gases of predominantly upper stratospheric photochemistry. ¿ 2000 American Geophysical Union