  A longest sequence $(v_1,\ldots,v_k)$ of vertices of a graph $G$ is a Grundy
total dominating sequence of $G$ if for all $i$, $N(v_i) \setminus
\bigcup_{j=1}^{i-1}N(v_j)\not=\emptyset$. The length $k$ of the sequence is
called the Grundy total domination number of $G$ and denoted
$\gamma_{gr}^{t}(G)$. In this paper, the Grundy total domination number is
studied on four standard graph products. For the direct product we show that
$\gamma_{gr}^t(G\times H) \geq \gamma_{gr}^t(G)\gamma_{gr}^t(H)$, conjecture
that the equality always holds, and prove the conjecture in several special
cases. For the lexicographic product we express $\gamma_{gr}^t(G\circ H)$ in
terms of related invariant of the factors and find some explicit formulas for
it. For the strong product, lower bounds on $\gamma_{gr}^t(G \boxtimes H)$ are
proved as well as upper bounds for products of paths and cycles. For the
Cartesian product we prove lower and upper bounds on the Grundy total
domination number when factors are paths or cycles.
