In the coming years, several cosmological surveys will rely on imaging data to estimate the redshift of galaxies, using traditional filter systems with 4-5 optical broad bands; narrower filters improve the spectral resolution, but strongly reduce the total system throughput. We explore how photometric redshift performance depends on the number of filters nf , characterizing the survey depth by the fraction of galaxies with unambiguous redshift estimates. For a combination of total exposure time and telescope imaging area of 270 hr m2, 4-5 filter systems perform significantly worse, both in completeness depth and precision, than systems with nf gsim 8 filters. Our results suggest that for low nf the color-redshift degeneracies overwhelm the improvements in photometric depth, and that even at higher nf the effective photometric redshift depth decreases much more slowly with filter width than naively expected from the reduction in the signal-to-noise ratio. Adding near-IR observations improves the performance of low-nf systems, but still the system which maximizes the photometric redshift completeness is formed by nine filters with logarithmically increasing bandwidth (constant resolution) and half-band overlap, reaching ~0.7 mag deeper, with 10% better redshift precision, than 4-5 filter systems. A system with 20 constant-width, nonoverlapping filters reaches only ~0.1 mag shallower than 4-5 filter systems, but has a precision almost three times better, δz = 0.014(1 + z) versus δz = 0.042(1 + z). We briefly discuss a practical implementation of such a photometric system: the ALHAMBRA Survey.