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This article is devoted to the global output‐feedback stabilization of nonlinear systems with unknown control directions and unmeasured states dependent growth whose rate is an unknown arbitrary function of output. In view of limited capability of continuous feedback in compensating system uncertainties and nonlinearities, a novel switching adaptive scheme is proposed. The scheme involves a logic‐based switching mechanism, based on which, a proper controller is selected (from a family of candidate controllers) to act on the system during its execution interval. The main differences between the proposed scheme and those in the related works are as follows. First, the set of candidate controllers is determined online, rather than specified in advance, which provides foundations for compensating arbitrary function‐of‐output growth rate. Second, an iterative function composed of system memory signals is introduced into the switching mechanism, aiming to evaluate the differences between the system states and observe states. Finally, when switching stops, only by introducing an extra transformation and adopting elaborate analysis, can the boundedness of the closed‐loop system signals be obtained, rather than those in the related works where the boundedness can be naturally established once switching stops. The constructed switching adaptive controller guarantees that all the signals of the resulting closed‐loop system are bounded and ultimately converge to zero. [ABSTRACT FROM AUTHOR]