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In this article we investigate some "unexpected" properties of the "InfinitePower Tower". \[y = f(x) = {x^{{x^{{x^{{x^{\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu  \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} \] The materialcollected here is also intended as a potential guide for teachers ofhigh-school/undergraduate students interested in planning an activity ofinvestigative mathematics in the classroom, where the knowledge is gainedthrough the active, creative and cooperative use of diversified mathematicaltools (and some ingenuity). The activity should possibly be carried on with alaboratorial style, with no preclusions on the paths chosen and undertaken bythe students and with little or no information imparted from the teacher'sdesk. The teacher should then act just as a guide and a facilitator. Themathematical requisites to follow this path are: functions, properties ofexponentials and logarithms, sequences, limits and derivatives. The topicspresented should then be accessible to undergraduate or "advanced high school"students.

The strange properties of the infinite power tower