  In this paper, we are mainly concerned with studying arbitrary unbounded
square roots of linear operators as well as some of their basic properties. The
paper contains many examples and counterexamples. As an illustration, we give
explicit everywhere defined unbounded non-closable $nth$ roots of the identity
operator as well as the zero operator. We also show a non-closable unbounded
operator without any non-closable square root. Among other consequences, we
have a way of finding everywhere defined bijective operators, everywhere
defined operators which are surjective without being injective and everywhere
defined operators which are injective without being surjective. Some related
results on nilpotence are also given.
