The smallest dimension is 2 : you may take the open annulus and Möbius band : their tangent bundles are both diffeomorphic to $S^1\times R^3$. I think you can obtain $k=\infty$ in three dimensions, by taking various Whitehead manifolds (contractible 3-manifolds not diffeomorphic to $R^3$ but whose product with $R$ is diffeomorphic to $R^4$, if I'm not mistaking)mistaking, see this question, answers and comments).
The smallest dimension is 2 : you may take the open annulus and Möbius band : their tangent bundles are both diffeomorphic to $S^1\times R^3$. I think you can obtain $k=\infty$ in three dimensions, by taking various Whitehead manifolds (contractible 3-manifolds not diffeomorphic to $R^3$ but whose product with $R$ is diffeomorphic to $R^4$, if I'm not mistaking).