We report the results of an intensive program of photometry of the old nova V603 Aquilae (= Nova Aquilae 1918). In four separate observing runs of high-speed photometry, totaling 15 nights, no evidence is found for previously reported signals with periods of 61.4 and 15.6 minutes, or any other coherent signal with a period in the range 20 seconds to 3 hours. This makes it very unlikely that the star contains a strongly magnetic white dwarf, as has been suggested. The light curves do consistently show a slow wave with a period of approximately 3.5 hours, about 5% longer than the accepted orbital period. This period is stable on time scales of at least a few weeks. Published observations are too scarce to determine a long-term ephemeris, but the period increased between 1981 and 1990 at a mean rate of approximately 5 seconds year-1. We discuss several models for the origin of the periodicity. There is some resemblance to the well-known "superhumps" of the SU Ursae Majoris class of dwarf novae. This suggests that the star may be stuck in a more or less permanent state of high mass transfer, with the photometric period arising from the precession of an eccentric accretion disk.