We present our analysis of the long-cadence Kepler data for the well-studied Blazhko star RR Lyr, gathered during the first two quarters of the satellite's observations and covering a total of 127 d. Besides being of great importance for our understanding of RR Lyrae stars in general, these RR Lyr data can be regarded as a case study for observations of bright stars with Kepler. Kepler can perform high-precision photometry on targets like RR Lyr, as the saturated flux is conserved to a very high degree. The Kepler data on RR Lyr are revolutionary in several respects. Even with long-cadence sampling (one measurement per 29.4 min), the unprecedented precision (< mmag) of the Kepler photometry allows the study of the star's extreme light-curve variations in detail. The multiplet structures at the main frequency and its harmonics, typical for Blazhko stars, are clearly detected up to the quintuplets. For the first time, photometric data of RR Lyr reveal the presence of half-integer frequencies, linked to a period-doubling effect. This phenomenon may be connected to the still unexplained Blazhko modulation. Moreover, with three observed Blazhko cycles at our disposal, we observe that there is no exact repetition in the light-curve changes from one modulation cycle to the next for RR Lyr. This may be due to additional periodicities in the star, or to transient or quasi-periodic changes.