For pure-feedback nonlinear systems under asymmetric output constraint, we present a low-cost neuroadaptive tracking control solution with salient features benefited from two design steps. In the first step, a novel output-dependent universal barrier function (ODUBF) is constructed such that not only the restrictive condition on constraining boundaries/functions is removed but also both constrained and unconstrained cases can be handled uniformly without the need for changing the control structure. In the second step, to reduce the computational burden caused by the neural network (NN)-based approximators, a single parameter estimator is developed so that the number of adaptive law is independent of the system order and the dimension of system parameters, making the control design inexpensive in computation. Furthermore, it is shown that all signals in the closed-loop system are semiglobally uniformly ultimately bounded, the tracking error converges to an adjustable neighborhood of the origin, and the violation of output constraint is prevented. The effectiveness of the proposed method can be validated via numerical simulation.