SignificanceEarth and Venus have significant atmospheres, but Mercury does not. Thousands of exoplanets are known, but we know almost nothing about rocky exoplanet atmospheres. Many rocky exoplanets were formed by a sub-Neptune-to-super-Earth conversion process during which planets lose most of their H2-rich (primary) atmospheres and are reduced in volume by a factor of >2. Does such a gas-rich adolescence increase or decrease the likelihood that super-Earths will subsequently exhibit a H2-poor (secondary) atmosphere? We show that secondary atmospheres exsolved from the magma ocean are unlikely to be retained by super-Earths, but it is possible for volcanic outgassing to revive super-Earth atmospheres. For M-dwarf planetary systems, super-Earths that have atmospheres close to the star likely were formed with abundant volatiles.