The sequestration of carbon dioxide (CO2) through storage into deep saline aquifers represents an indispensable support technology to achieve the zero-carbon target necessary to mitigate the impact of CO2 on climate change. The effectiveness of the sequestration process, partly driven by the convective dissolution of CO2 in brine, is nowadays well characterized for two-dimensional geometries, low permeabilities, and small pressures of injection of CO2. However, reliable predictions of process-efficiency are missing because of the lack of full understanding of the three-dimensional (3D) spatio-temporal behaviour of CO2-rich convective fingers in brine over a large range of injection pressures. Here, we show that the convective dissolution is determined by the instability of the boundary layer formed at the interface between the two phases and is totally independent of the overall vertical size. Experiments were conducted over a broad range of injection pressures, close to process-relevant conditions. The results show the formation of complex 3D structures, including interconnecting stream tubes at the CO2-liquid interface, which could not be detected in previous 2D Hele-Shaw studies, and fingerings. A scale-free theoretical modelling of the convective process allows us to remap our laboratory results to length-scales of relevance for geological reservoirs. The experiments and the model show that the times needed for the onset of convection and the convective flux are independent of the system size.