  Rydberg excitons (analogues of Rydberg atoms in condensed matter systems) are
highly excited bound electron-hole states with large Bohr radii. The
interaction between them as well as exciton coupling to light may lead to
strong optical nonlinearity, with applications in sensing and quantum
information processing. Here, we achieve strong effective photon-photon
interactions (Kerr-like optical nonlinearity) via the Rydberg blockade
phenomenon and the hybridisation of excitons and photons forming polaritons in
a Cu$_2$O-filled microresonators. Under pulsed resonant excitation polariton
resonance frequencies are renormalised due to the reduction of the
photon-exciton coupling with increasing exciton density. Theoretical analysis
shows that the Rydberg blockade plays a major role in the experimentally
observed scaling of the polariton nonlinearity coefficient as $\propto n^{4.4
\pm 1.8}$ for principal quantum numbers up to n = 7. Such high principal
quantum numbers studied in a polariton system for the first time are essential
for realisation of high Rydberg optical nonlinearities, which paves the way
towards quantum optical applications and fundamental studies of
strongly-correlated photonic (polaritonic) states in a solid state system.
