The Type Ia supernova (SN Ia) LSQ14fmg exhibits exaggerated properties that may help to reveal the origin of the "super-Chandrasekhar"(or 03fg-like) group. The optical spectrum is typical of a 03fg-like SN Ia, but the light curves are unlike those of any SNe Ia observed. The light curves of LSQ14fmg rise extremely slowly. At -23 rest-frame days relative to B-band maximum, LSQ14fmg is already brighter than MV = -19 mag before host extinction correction. The observed color curves show a flat evolution from the earliest observation to approximately 1 week after maximum. The near-infrared light curves peak brighter than -20.5 mag in the J and H bands, far more luminous than any 03fg-like SNe Ia with near-infrared observations. At 1 month past maximum, the optical light curves decline rapidly. The early, slow rise and flat color evolution are interpreted to result from an additional excess flux from a power source other than the radioactive decay of the synthesized 56Ni. The excess flux matches the interaction with a typical superwind of an asymptotic giant branch (AGB) star in density structure, mass-loss rate, and duration. The rapid decline starting at around 1 month past B-band maximum may be an indication of rapid cooling by active carbon monoxide (CO) formation, which requires a low-temperature and high-density environment. These peculiarities point to an AGB progenitor near the end of its evolution and the core degenerate scenario as the likely explosion mechanism for LSQ14fmg.