Besides purely academic interest, giant field enhancement within subwavelength particles at light scattering of a plane electromagnetic wave is important for numerous applications ranging from telecommunications to medicine and biology. In this paper, we experimentally demonstrate the enhancement of the intensity of the magnetic field in a high-index dielectric cylinder at the proximity of the dipolar Mie resonances by more than two orders of magnitude for both the TE and TM polarizations of the incident wave. We present a complete theoretical explanation of the effect and show that the phenomenon is very general - it should be observed for any high-index particles. The results explain the huge enhancement of nonlinear effects observed recently in optics, suggesting a new landscape for all-dielectric nonlinear nanoscale photonics.