Context. The central star of the HR 8799 system is a γ Doradus-type pulsator. The system harbours four planetary-mass companions detected by direct imaging, and is a good solar system analogue. The masses of the companions are not accurately known because the estimation depends greatly on the age of the system, which is also not known with sufficient accuracy. Asteroseismic studies of the star might help to better constrain the age of HR 8799. We organized an extensive photometric and multi-site spectroscopic observing campaign to study the pulsations of the central star. Aims. The aim of the present study is to investigate the pulsation properties of HR 8799 in detail via the ultra-precise 47 d nearly continuous photometry obtained with the Microvariability and Oscillations in STars (MOST) space telescope, and to find as many independent pulsation modes as possible, which is the prerequisite for an asteroseismic age determination. Methods. We carried out Fourier analysis of the wide-band photometric time series. Results. We find that resonance and sudden amplitude changes characterize the pulsation of HR 8799. The dominant frequency is always at f 1 = 1.978 d −1 . Many multiples of one-ninth of the dominant frequency appear in the Fourier spectrum of the MOST data: n/9 f 1 , where n = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 17, 18}. Our analysis also reveals that many of these peaks show strong amplitude decrease and phase variations even on the 47 d time scale. The dependencies between the pulsation frequencies of HR 8799 make the planned subsequent asteroseismic analysis rather difficult. We point out some resemblance between the light curve of HR 8799 and the modulated pulsation light curves of Blazhko RR Lyrae stars.