  We study polynomial identities of algebras with involution of nonassociative
algebras over a field of characteristic zero. We prove that the growth of the
sequence of $*$-codimensions of a finite-dimensional algebra is exponentially
bounded. We construct a series of finite-dimensional algebras with fractional
$*$-PI-exponent. We also construct a family of infinite-dimensional algebras
$C_\alpha$ such that ${\rm exp}^*(C_\alpha)$ does not exist.
