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Crustal magma chambers can grow to be hundreds to thousands of cubic kilometers, potentially feeding catastrophic caldera‐forming eruptions. Smaller volume chambers are expected to erupt frequently and freeze quickly; a major outstanding question is how magma chambers ever grow to the sizes required to sustain the largest eruptions on Earth. We use a thermo‐mechanical model to investigate the primary factors that govern the extrusive:intrusive ratio in a chamber, and how this relates to eruption frequency, eruption size, and long‐term chamber growth. The model consists of three fundamental timescales: the magma injection timescale τin, the cooling timescale τcool, and the timescale for viscous relaxation of the crust τrelax. We estimate these timescales using geologic and geophysical data from four volcanoes (Laguna del Maule, Campi Flegrei, Santorini, and Aso) to compare them with the model. In each of these systems, τin is much shorter than τcool and slightly shorter than τrelax, conditions that in the model are associated with efficient chamber growth and simultaneous eruption. In addition, the model suggests that the magma chambers underlying these volcanoes are growing at rates between ~10−4 and 10−2 km3/year, speeding up over time as the chamber volume increases. We find scaling relationships for eruption frequency and size that suggest that as chambers grow and volatiles exsolve, eruption frequency decreases but eruption size increases. These scaling relationships provide a good match to the eruptive history from the natural systems, suggesting that the relationships can be used to constrain chamber growth rates and volatile saturation state from the eruptive history alone. Plain Language Summary: Magma chambers in the Earth's crust grow by incremental addition of new magma from deeper reservoirs, and over time can reach volumes that would fill the entire Grand Canyon. However, small magma chambers in the earliest stages of formation are prone to frequent eruptions and will lose heat quickly to the surrounding crust, both of which supposedly impede growth. Therefore, an important question is how magma chambers can possibly grow to such large sizes. Here we present results of physics‐based modeling aimed at understanding what conditions allow magma chambers to grow. We test effects of chamber size, rate of magma supply, water content in the magma, and plasticity of the crust hosting the chamber. Results suggest that growth is promoted when chambers cool slowly and are hosted within pliable crust that can easily relax pressures that build within the chamber. Surprisingly, we found that for a particular range of crustal pliability, growth is accompanied by frequent volcanic eruptions. We compared these results to four large volcanoes in Chile, Italy, Greece, and Japan. Model predictions for eruption frequency and chamber growth rates are a good match to what we observe at these volcanoes from the rock record and active monitoring systems. Key Points: Coupled thermo‐mechanical modeling suggests that magma chambers grow when recharge and viscous relaxation of the crust are fast compared to chamber coolingMagma chambers containing exsolved volatiles may shrink over time despite constant recharge because high magma compressibility leads to large eruption volumesEruptive phases at Laguna del Maule, Campi Flegrei, Santorini, and Aso reflect growth of chambers; amount of growth can be estimated from eruption frequency and size [ABSTRACT FROM AUTHOR]