We present observations of SN 2015bn (=PS15ae = CSS141223-113342+004332 = MLS150211-113342+004333), a Type I superluminous supernova (SLSN) at redshift z = 0.1136. As well as being one of the closest SLSNe I yet discovered, it is intrinsically brighter (MU ≈ -23.1) and in a fainter galaxy (MB ≈ -16.0) than other SLSNe at . We used this opportunity to collect the most extensive data set for any SLSN I to date, including densely sampled spectroscopy and photometry, from the UV to the NIR, spanning -50 to +250 days from optical maximum. SN 2015bn fades slowly, but exhibits surprising undulations in the light curve on a timescale of 30-50 days, especially in the UV. The spectrum shows extraordinarily slow evolution except for a rapid transformation between +7 and +20-30 days. No narrow emission lines from slow-moving material are observed at any phase. We derive physical properties including the bolometric luminosity, and find slow velocity evolution and non-monotonic temperature and radial evolution. A deep radio limit rules out a healthy off-axis gamma-ray burst, and places constraints on the pre-explosion mass loss. The data can be consistently explained by a ≳ 10 M⊙ stripped progenitor exploding with erg kinetic energy, forming a magnetar with a spin-down timescale of ∼20 days (thus avoiding a gamma-ray burst) that reheats the ejecta and drives ionization fronts. The most likely alternative scenario - interaction with ∼20 M⊙ of dense, inhomogeneous circumstellar material - can be tested with continuing radio follow-up.