The author studies the growth function \(c_n(\mathbf V)\) for a variety of Lie algebras \(\mathbf V\), where \(c_n(\mathbf V)\) is the dimension of the linear hull of the set of multilinear words with \(n\) different letters in the free algebra of the variety \(\mathbf V\). For every non-trivial variety of linear algebras \(\mathbf V\), some complexity function \({\mathcal C}(\mathbf V,z)\) is constructed which is an entire function of a complex variable. Namely, \({\mathcal C}(\mathbf V,z)= \sum_{n=1}^{\infty} c_n(\mathbf V)z^n/n!\), \(z\) is a complex variable. In the case of a polynilpotent variety \(\mathbf V\) of Lie algebras, an estimate for a function \({\mathcal C}(\mathbf V,z)\) is obtained. In most cases it is of infinite order. An asymptotic for the function \(c_n(\mathbf V)\) is obtained for a polynilpotent variety \(\mathbf V\). The author also proves an analogue of Regev’s theorem for Lie algebras on upper estimates for the growth of arbitrary varieties.