We propose an adaptive output-feedback controller for a general class of nonlinear triangular (strictfeedback-like) systems. The design is based on our recent results on a dual high-gain observer and controller architecture with a dynamic scaling. The technique provides strong robustness properties and allows the system class to contain unknown functions dependent on all states and involving unknown parameters (with no magnitude bounds required). Unlike our earlier result on this problem where a time-varying design of the high-gain scaling parameter was utilized, the technique proposed here achieves an autonomous dynamic controller by introducing a novel design of the observer, the scaling parameter, and the adaptation parameter. This provides a time-invariant dynamic output-feedback controller for the benchmark open problem proposed in our earlier work with no magnitude bounds or sign information on the unknown parameter being necessary.