Another good $2$-dimensional example is torus punctured once and sphere punctured three times. These become diffeomorphic when crossed with $\mathbb R$, and they have trivial tangent bundles. More generally, if you puncture a closed genus $g$ surface $n>0$ times then the product with $\mathbb R$ depends only on $2g+n$, so in dimension $2$ you can get arbitrarily large finite $k$.

But maybe you wanted a compact (without boundary) example?