Identities of unitary finite-dimensional algebras

We deal with growth functions of sequences of codimensions of identities in finite-dimensional algebras with unity over a field of characteristic zero. For three-dimensional algebras, it is proved that the codimension sequence grows asymptotically as $a^n$, where $a$ is $1,2$, or $3$. For arbitrary finite-dimensional algebras, it is shown that the codimension growth either is polynomial or is not slower than $2^n$. We give an example of a finite-dimensional algebra with growth rate $a^n$ with fractional exponent $a=\frac3{\sqrt[3]4}+1$.