Let \(F\) be a field of characteristic zero and let \(A\) be a (not necessarily associative) \(F\)-algebra. By analogy with associative PI-algebras, one introduces the sequence \(c_n(A)\) of codimensions of the multilinear polynomial identities of \(A\), the \(S_n\)-cocharacter \(\chi_n(A)=\sum_{\lambda\vdash n}m_\lambda\chi_\lambda\), and the colength \(l_n(A)=\sum_{\lambda\vdash n}m_\lambda\). A valuable information on the combinatorial and structural properties of \(A\) can be obtained from the asymptotic behaviour of \(c_n(A)\), \(\chi_n(A)\), \(l_n(A)\). The authors show that for finite dimensional nonassociative algebras the colength sequence of \(A\) is polynomially bounded and the codimension sequence cannot have intermediate growth. Then the authors construct a series of examples of nonassociative algebras with intermediate growth of the codimensions. They show that for any real number \(0<\beta<1\), there exists an algebra \(A\) whose sequence of codimensions grows like \(n^{n^\beta}\).