The formation of soap bubbles from thin films is accompanied by topological transitions. Here we show how a magnetic topological structure, a skyrmion bubble, can be generated in a solid-state system in a similar manner. Using an inhomogeneous in-plane current in a system with broken inversion symmetry, we experimentally "blow" magnetic skyrmion bubbles from a geometrical constriction. The presence of a spatially divergent spin-orbit torque gives rise to instabilities of the magnetic domain structures that are reminiscent of Rayleigh-Plateau instabilities in fluid flows. We determine a phase diagram for skyrmion formation and reveal the efficient manipulation of these dynamically created skyrmions, including depinning and motion. The demonstrated current-driven transformation from stripe domains to magnetic skyrmion bubbles could lead to progress in skyrmion-based spintronics.