This book is an outgrowth of a seemingly trivial question, namely: what happens with the Feigenbaum universality when one restricts attention to functions of |x|^2N (instead of x^2), with N very large? Closer numerical investigation [26,29,35,40] reveals that this question is hard, and that some aspects of the problem must become singular in the limit as N->(infinity). In this work, we study this singular limit, and sketch its relation to the case of finite, but large N.