Anomalously large low temperature phonon anharmonicities can lead to static as well as dynamical changes in the low temperature properties of the vibrational system. In this work, we focus our attention on the low temperature lattice anharmonicity and its effect on the electron-phonon ground state. We are, in particular, motivated by certain high temperature superconductors. The third and fourth order anharmonic coupling constants for YBCO, LBCO and several other superconducting compounds are extracted from their measured elastic constants using the anharmonic elastic continuum model. The coupling constants are then used to extract the average anharmonic potential energy for the transverse and longitudinal modes in the Cu-0 planes. We find that, anharmonic contribution to the lattice potential relative to the harmonic one is unusually high for all examined high Tc compounds. The presence of anharmonic phonons elicits non-perturbative dynamical effects in the ground state of the electron-phonon system. Phonon correlations induced by anharmonic effects enhance the electron-phonon interaction which then create a self-consistent mechanism to act back on the ground state of the electron-phonón system. In result, strong momentum correlations are created and the ground state comprises fluctuating polarons. The zero point fluctuations and other ground state properties are obtained by self-consistent numerical calculations. The influence of low temperature phonon anharmonicity on the superconducting properties in the intermediate coupling range is also investigated. It is shown that, the otherwise bare electron-phonon coupling is strengthened in the presence of correlated polarons and the zero point fluctuations are enhanced. Within this frame, it is plausible to achieve superconducting transition temperatures as high as %20 of the characteristic vibrational energy scale. The non-perturbative, self-consistent formalism thus introduced also offers an account for the recently observed temperature anomalies near Tc in the Debye-Waller factor and dynamical pair correlations of certain high temperature superconductors.