Variant of TSP: allow each vertex to be visited at most twice
Let $T = (V,d,k)$ be an instance of Metric-TSP, where $V$ is a vertex set, $d: V\times V \rightarrow \mathbb{R}_{\geq 0}$ is a distance with triangle inequality, and $k\in \mathbb{R}_{\geq 0}$.

Suppose that $T$ is a positive instance of Metric-TSP (i.e. there exists a tour with total length $\leq k$. Then $T$ is also a positive instance for your variant (just take the same tour).

Suppose $T$ is a positive instance of your variant, and let $t = (v_1, v_2,\ldots,v_r)$ denote a "tour" with total length $\leq k$. Whenever there is a vertex $v$ which appears twice in $t$ one can safely delete one of the two appearances as every vertex will still be visited and by the triangle inequality the total length of $t$ can only decrease. Once you have deleted all the duplicates, you are thus left with a tour of length $\leq k$, which shows that $T$ is also a positive instance of Metric-TSP.

Thus, there is a trivial reduction (namely the identity) from Metric-TSP to your variant. As Metric-TSP is NP-complete and your variant is in NP, your variant is also NP-Complete.