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Tuning the solubility of fuzzy polymer microgels by external triggers, such as temperature or pH, provides a unique mechanism for controlling the porosity and size of colloidal particles on the nanoscale. As a consequence, these smart microgel particles are being considered for applications ranging from viscosity modifiers and sensing to drug delivery and as models for the glass and the jamming transition. Despite their widespread use, little is known about how these soft particles adapt their shape and size under strong mechanical compression. We use a combination of precise labeling protocols and two-color superresolution microscopy to unravel the behavior of tracer microgels inside densely packed soft solids. We find that interpenetration and shape deformation are dominant until, in the highly overpacked state, this mechanism saturates and the only remaining way to further densify the system is by isotropic compression. [ABSTRACT FROM AUTHOR]