We present the first limits on the Epoch of Reionization 21 cm H I power spectra, in the redshift range z = 7.910.6, using the Low-Frequency Array (LOFAR) High-Band Antenna (HBA). In total, 13.0 hr of data were used from observations centered on the North Celestial Pole. After subtraction of the sky model and the noise bias, we detect a non-zero Delta(2)(I)=(56 +/- 13 mK)(2) (1-sigma) excess variance and a best 2-sigma upper limit of Delta(2)(21) < (79.6 mK)(2) at k = 0.053 h cMpc(-1) in the range z = 9.610.6. The excess variance decreases when optimizing the smoothness of the direction- and frequency-dependent gain calibration, and with increasing the completeness of the sky model. It is likely caused by (i) residual side-lobe noise on calibration baselines, (ii) leverage due to nonlinear effects, (iii) noise and ionosphere-induced gain errors, or a combination thereof. Further analyses of the excess variance will be discussed in forthcoming publications.