We present a curious effect observed during the dissolution process of water-immersed long-chain alcohol drops with an entrapped air bubble. These droplets dissolve while entrapping an air bubble pinned at the substrate. We qualitatively describe and explain four different phases that are found during the dissolution of this kind of system. The dissolution rate in the four phases differ dramatically. When the drop-water interface and the air bubble contact each other, rapid cyclic changes of the morphology are found: The breakage of the thin alcohol layer in between the bubble and the water leads to the formation of a three phase contact line. If the surface tension of the water-air interface supersedes those of the alcohol-water and alcohol-air interfaces, alcohol from the droplet is pulled upwards, leading to a closure of the air-water interface and the formation of a new thin alcohol film, which then dissolves again, leading to a repetition of the series of events. We call this sequence of events Marangoni puffing. This only happen for alcohols of appropriate surface tension. The Marangoni puffing is an intermediate state. In the final dissolution phases the Marangoni forces dramatically accelerate the dissolution rate, which then becomes one order of magnitude faster than the purely buoyancy-convective driven dissolution. Our results have bearing on various dissolution processes in multicomponent droplet systems.