Check the paper Antonelli, P. L.; Burghelea, D.; Kahn, P. J.
[*The non-finite homotopy type of some diffeomorphism groups.*][1] 
Topology 11 (1972), 1–49. 

There they mention an old result of Palais (*[Homotopy theory of infinite dimensional manifolds][2]*. Topology 5 (1966),  1-16) which states that  the identity component of ${\rm Diff}_0(M)$ has the homotopy type of a *countable* $CW$-complex.

In their paper, Antonelli, Burghelea and Kahn prove that for many smooth manifolds (including spheres of dimension $\geq 7$) the group ${\rm Diff}_0(M)$ does not have the homotopy type of a *finite* $CW$-complex. (This is highly nontrivial.)  

Above, by diffeomorphisms they mean smooth diffeomorphisms and the topology is the $C^\infty$-topology.


  [1]: http://www.sciencedirect.com/science/article/pii/0040938372900213
  [2]: http://www.sciencedirect.com/science/article/pii/0040938366900024