It makes no sense to take a single x^x from the infinite complex
Because it is anything but a standard function like x^a or a^x. So any discussion on x^x in terms of the properties of the well-known functions is irrelevant. Anyway, the complex structure should be treated as a whole, do not try to analyze it. Otherwise, any outcomes you think having acquired will get you misled. Seems we are not on the same page yet on the monster structure.