For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V(G) such that |N[v] ∩S| ≥k for every vertex v, where N[v] = {v} ∪{u ∈V(G):uv ∈ E(G)}. The upper k-tuple domination number of G, denoted by Γ_×k(G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ_×k(G) for r-regular graphs G with r ≥k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ_×2(G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs.