Optical microscopy of single bacteria growing on solid agarose support is a powerful method for studying the natural heterogeneity in growth and gene expression. While the material properties of agarose make it an excellent substrate for such studies, the sheer number of exponentially growing cells eventually overwhelms the agarose pad, which fundamentally limits the duration and the throughput of measurements. Here we overcome the limitations of exponential growth by patterning agarose pads on the sub-micron-scale. Linear tracks constrain the growth of bacteria into a high density array of linear micro-colonies. Buffer flow through microfluidic lines washes away excess cells and delivers fresh nutrient buffer. Densely patterned tracks allow us to cultivate and image hundreds of thousands of cells on a single agarose pad over 30–40 generations, which drastically increases single-cell measurement throughput. In addition, we show that patterned agarose can facilitate single-cell measurements within bacterial communities. As a proof-of-principle, we study a community of E. coli auxotrophs that can complement the amino acid deficiencies of one another. We find that the growth rate of colonies of one strain decreases sharply with the distance to colonies of the complementary strain over distances of only a few cell lengths. Because patterned agarose pads maintain cells in a chemostatic environment in which every cell can be imaged, we term our device the single-cell chemostat. High-throughput measurements of single cells growing chemostatically should greatly facilitate the study of a variety of microbial behaviours.