Finite dimensional Lie groups have the nice property that if $V$ is a small neighborhood of the identity, and $U \subset V$ another neighborhood, then $V$ is covered by $U^k$ (the set of all products of $k$ elements of $U$) for some $k$. This follows from the fact that V is relatively compact. 


Does the same property hold for small enough neighborhoods of the identity in an infinite dimensional Lie group? My favorite example is the group of compactly supported diffeomorphisms of $\mathbb{R}^n$ -- is it known whether it has this property? 


Remark: I'm pretty sure this property is not equivalent to local compactness, so the fact that $Diff\_c(\mathbb{R}^n)$ is not locally compact doesn't give an answer. 



Neighborhoods of the identity in diffeomorphism groups
What helps a little is that for any (second countable connected) smooth manifold $M$, the connected component $Diff\_c(M)\_0$ of the group $Diff\_c(M)$ is perfect (Epstein) and simple (Thurston). On the other hand, $\exp: \mathfrak X\_c(M)\to Diff\_c(M)$ is far from being locally surjective. Grabowski has shown, that for $\dim(M)\ge 2$ there exists a smooth curve in $Diff\_c(M)$ starting at the identity such that the element of these curve (sauf the identity) form a free generating system of a free subgroup of $Diff\_c(M)$ such that no element $\ne 1$ of this free subgroup embeds into a flow. This free subgroup of uncountably many generators is contractible to 1 (slide each generator to 1 along the curve). 


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Some references: 


* (1) Martins Bruveris, François-Xavier Vialard: On Completeness of Groups of Diffeomorphisms. arXiv:1403.2089
* (2) S. Haller and T. Rybicki and J. Teichmann: Smooth perfectness for the group of diffeomorphisms. J. Geom. Mech. 5(2013), 281–294. Preprint: arXiv:math/0409605
* (3) David Mumford, Peter W. Michor: On Euler's equation and `EPDiff'. Journal of Geometric Mechanics 5, 3 (2013), 319-344
[(pdf)](http://www.mat.univie.ac.at/~michor/eulerlink.pdf)
* (4) M. Bauer, J. Escher and B. Kolev: Local and Global Well-posedness of the fractional order EPDiff equation on R^d. To appear in Journal of Differential equations.


In [3], Thm 2, see (1) and (4) for more information, it is shown that the right invariant Sobolev metric of integral order $s>\frac{n+3}2$ is geodesically complete on $Diff\_{H^\infty}(\mathbb R^n)$ and the corresponding Riemannian exponential mapping is a diffeomorphism from a suitable $H^s$-ball in $\mathfrak X\_{H^\infty}(\mathbb R^n)$ into the group. 
This easily implies the result that the OP asked, but only for $H^s$-balls; thus for the Sobolev completion, the topological group (with smooth right translations) 
$$Diff\_{H^s}(\mathbb R^n) = \{Id + f: f\in H^s(\mathbb R^n)^n, \det(Id + df) > 0\}.$$
Namely, consider any $H^s$-ball $V$ of radius $r$, Then any open neighborhood in $V$ contains a $H^s$-ball of radius $\epsilon>0$. Papers (1) and (4) show that one can connect any diffeomorphism $\phi\in V$ to the identity by a geodesic $c$ of length $<r$. Cut this geodesic in $\frac{\epsilon}2$-pieces $\{c(t): t\_i\le t\le t\_{i+1}\}$, $0=t\_0<t\_1<t\_2<\dots<t\_N=r$.
Then $c\_1(t)=\{c(t): t\_1\le t\le r\}. c(t\_1)^{-1}$ has length $r-t\_1$, and finally 
$\phi=c(r)= c\_N(r).c\_{N-1}(t\_{N-1}).\dots.c\_1(t\_2).c(t\_1)$. 


Moreover, for $Diff\_{H^\infty}(\mathbb R^n)$, where $H^\infty$ is the intersection of all Sobolev spaces, a Frechet space, the result also holds, since each neighborhood of the identity in $Diff\_{H^\infty}(\mathbb R^n)$ contains a $H^k$-neigbourhood for sufficiently large but finite $k$, and we can apply the result with the $H^k$-metric.


But $Diff\_c(\mathbb R^n)$ has many more open neighborhoods, and for these nothing follows. 


