Given a smooth manifold $M$, we define the differentiable structure on $TM$ in the usual way.

I would like to know examples of two smooth manifolds which are non-diffeomorphic, but their tangent bundles are.

Which is the smallest dimension in which one can find such examples?

What if I ask the same question for $k$ pairwise non-diffeomorphic manifolds?

Can we have $k=\infty$?