Towers of fields F_1 is contained in F_2 is contained in F_3 are considered, where F_3/F_2 is a quadratic extension and F_2/F_1 is an extension, which is either quadratic or of odd degree or purely transcendental of degree 1. Numerous examples of the above types such that the extension F_3/F_1 is not 4-excellent are constructed. Also it is shown that if k is a field, char k ≠ 2, and l/k is an arbitrary field extension of fourth degree, then there exists a field extension F/k such that the fourth degree extension lF/F is not 4-excellent.
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