Commenting on Peter Michor's answer, I want to say that we can however use the exponential map.

*Is it true* that the image of the exponential map is locally dense near the origin? If so, let us take open neighbourhoods $U\subset V$ of the identity such that $\overline{\exp(Vect_c(M))}\supset \overline{V}$. Given any $g\in V$, we can find $h_0\in U$ arbitrarily close to $id$ so that $gh_0^{-1}=f$ is the time $1$ map of a flow $\{f_t\}_{t\in[0,1]}$. Let $k=k(f)$ be the minimum integer such that $f_{1/k}$ belongs to $U$. Then $g$ is the product of $k+1$ diffeomorphisms in $U$:
$$g=f_{1/k}\circ\cdots\circ f_{1/k}\circ h_0.$$

*The problem* here is that $\overline{U}$ is not compact so it doesn't seem clear that we can have a uniform bound on the choice of $k$ (though we still have some freedom in the choice of $h_0$).

**Edit** Katie Mann is true that the image of the exponential map can happen to be not dense about the origin. There is one case for which I know that my question has a negative answer and this is a result by Nancy Kopell in Commuting diffeomorphisms, *Global Analysis, Proc. Symmpos. Pure Math.* vol **XIV** (1968), 165-184, see also Yoccoz, Petits diviseurs en dimension 1 *Astérisque* **231**, SMF (1995)):

Kopell's Theorem states that the $C^1$ centralizer is trivial for an open dense set of $C^1$ circle diffeomorphisms. In particular there exists an open dense set of diffeomorphisms of the circle which are not the time 1 map of a flow.