  Consider two graphs $G$ and $H$. Let $H^k[G]$ be the lexicographic product of
$H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by
itself $k$ times. In this paper, we determine the spectrum of $H^k[G]$ and
$H^k$ when $G$ and $H$ are regular and the Laplacian spectrum of $H^k[G]$ and
$H^k$ for $G$ and $H$ arbitrary. Particular emphasis is given to the least
eigenvalue of the adjacency matrix in the case of lexicographic powers of
regular graphs, and to the algebraic connectivity and the largest Laplacian
eigenvalues in the case of lexicographic powers of arbitrary graphs. This
approach allows the determination of the spectrum (in case of regular graphs)
and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example,
the spectrum of the lexicographic power of the Petersen graph with the googol
number (that is, $10^{100}$) of vertices is determined. The paper finish with
the extension of some well known spectral and combinatorial invariant
properties of graphs to its lexicographic powers.
