Give me an example of a topological manifold which is not a smooth manifold.

Theorem. Any topological manifold of dimension $2$ or $3$ can be endowed with a real analytic manifold structure. Furthermore, two such structures which are homeomorphic are $C^\omega$-diffeomorphic.

Proof. See Geometric topology in dimension $2$ and $3$ by E. Moise. $\Box$

Therefore, an example must have dimension at least $4$, such a construction is not easy, see for example the article A manifold which does not admit any differentiable structure by M. Kervaire.

You can see the example of Kervaire The "Easiest" non-smoothable manifold