The problem of state estimation for a system of coupled hyperbolic PDEs and ODEs with Lipschitz nonlinearities with boundary measurements is considered. An infinite dimensional observer with a linear boundary injection term is used to solve the state estimation problem. The interconnection of the observer and the system is written in estimation error coordinates and analyzed as an abstract dynamical system. The observer is designed to achieve global exponential stability of estimation error with respect to a suitable norm. Sufficient conditions in the form of matrix inequalities are proposed to design the observer. Numerical simulations support and corroborate the theoretical results.