In the present work, we are presenting computational simulation results for the flow in a human right internal carotid artery, exhibiting two saccular aneurysms close to each other. We utilize computer tomography data in order to extract a realistic geometric description of the region of interest. Aspects of the flow inside the aneurysms are discussed in connection to secondary motion patterns and inflow-outflow regimes. We construct residence time maps that exhibit strong non-uniformity, connected to the existence of fluid entering only the first, only the second, or both aneurysms. Preliminary evidence that the inflow-outflow patterns of the two aneurysms may be leading to particularly complex flow and to chaotic mixing is discussed, based on the apparent properties of both the residence time map iso-contours and the basins of attraction of the two aneurysms. Particular attention is paid in establishing grid independence for the computed results and for this reason a second order spatial discretization scheme is utilized, with resolutions ranging from approximately 110,000 to 1,070,000 tetrahedra.