  Let $G=(V,E)$ be a simple graph. For any integer $k\geq 1$, a subset of $V$
is called a $k$-tuple total dominating set of $G$ if every vertex in $V$ has at
least $k$ neighbors in the set. The minimum cardinality of a minimal $k$-tuple
total dominating set of $G$ is called the $k$-tuple total domination number of
$G$. In this paper, we introduce the concept of upper $k$-tuple total
domination number of $G$ as the maximum cardinality of a minimal $k$-tuple
total dominating set of $G$, and study the problem of finding a minimal
$k$-tuple total dominating set of maximum cardinality on several classes of
graphs, as well as finding general bounds and characterizations. Also, we find
some results on the upper $k$-tuple total domination number of the Cartesian
and cross product graphs.
