Chemical reaction chains with feedback of one of the products on the source of the chain are considered. A strategy is presented in terms of ordinary differential equations which creates one, two, and three positive Lyapunov exponents as the finite dimension of the system is increased. In particular, a nonlinear inhibition loop in a chemical reaction sequence controls the type of chaos. The bifurcation scenarios are studied and chaos and hyperchaos are found for broad regions of bifurcation parameter. Some implications for the occurrence of higher chaos in real systems are discussed.