We investigate mapping properties for the Bargmann transform on an extended family ofmodulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We provethat this transform is isometric and bijective from modulation spacesto convenient Lebesgue spaces of analytic functions. We use this to prove that suchmodulation spaces fulfill most of the continuity properties which are valid for modulationspaces with moderate weights. Finally we use the results to establish continuity propertiesof Toeplitz and pseudo-differential operators on these modulation spaces, and onGelfand-Shilov spaces.