  We construct a family of unital non-associative algebras $\{T_\alpha\vert~
2<\alpha\in\mathbb R\}$ such that $\underline{exp}(T_\alpha)=2$, whereas
$\alpha\le\overline{exp}(T_\alpha)\le\alpha+1$. In particular, it follows that
ordinary PI-exponent of codimension growth of algebra $T_\alpha$ does not exist
for any $\alpha> 2$. This is the first example of a unital algebra whose
PI-exponent does not exist.
