We report a luminous Type II supernova, ASASSN-15nx, with a peak luminosity of M-v = -20 mag that is between those of typical core-collapse supernovae and super-luminous supernovae. The post-peak optical light curves show a long, linear decline with a steep slope of 2.5 mag (100 day)(-1) (i.e., an exponential decline in flux) through the end of observations at phase approximate to 260 day. In contrast, the light curves of hydrogen-rich supernovae (SNe II-P/L) always show breaks in their light curves at phase similar to 100 day, before settling onto Co-56 radioactive decay tails with a decline rate of about 1 mag (100 day)(-1). The spectra of ASASSN-15nx do not exhibit the narrow emission-line features characteristic of Type IIn SNe, which can have a wide variety of light-curve shapes usually attributed to strong interactions with a dense circumstellar medium (CSM). ASASSN-15nx has a number of spectroscopic peculiarities, including a relatively weak and triangular-shaped H alpha emission profile with no absorption component. The physical origin of these peculiarities is unclear, but the long and linear post-peak light curve without a break suggests a single dominant powering mechanism. Decay of a large amount of Ni-56 (M-Ni = 1.6 +/- 0.2 M-circle dot) can power the light curve of ASASSN-15nx, and the steep light-curve slope requires substantial gamma-ray escape from the ejecta, which is possible given a low-mass hydrogen envelope for the progenitor. Another possibility is strong CSM interactions powering the light curve, but the CSM needs to be sculpted to produce the unique light-curve shape and avoid producing SN IIn-like narrow emission lines.