Finite-dimensional subgroups of circle diffeomorphism group Is there a sequence of connected finite-dimensional subgroups Gi of the circle diffeomorphism group G with the following properities:
(a) Gi is contained in Gj for i < j
(b) The union of Gi is dense in G
More rigorously "finite dimensional subgroup of circle diffeomorphism group" means a Lie group H with smooth faithful action on the circle.
In order to make sense of property (b) I have to specify a topology on G. I suspect that all reasonable topologies will yield the same answer, but for the sake of definiteness let's use the 
"sup-norm" topology. That is, given two diffeomorphism g1 and g2, I define the distance d(g1, g2) as 
supremum over x in S1 of d(g1(x), g2(x))
Here the latter d is the usual distance on the circle.
This is a metric and it induces a topology.
I suspect that the answer to my question is "no". Moreover, I suspect that there is no H as above with dimension > 3. But I might be wrong...
 A: If $G$ is a connected Lie group acting transitively and faithfully on a connected smooth $1$-manifold, then $G$ is at most $3$-dimensional; in fact its Lie algebra embeds in that of $SL_2(\mathbb R)$. (Edit: Alex Eskin's answer says this in detail, with a reference.)
Each orbit of an action of a connected topological group on a $1$-manifold is either open or one point. Thus if $G_i$ has only one non-fixed orbit then $G_i$ is at most $3$-dimensional.
Let $F_i\subset S^1$ be the set of points fixed by the action of $G_i$. The intersection of all the $F_i$ is the set of points fixed by the union of the $G_i$, so surely it is empty if the union of the $G_i$ is dense. By compactness it follows that for all big enough $i$ $F_i$ is empty, making $G_i$ at most $3$-dimensional.
A: The answer is indeed no, as described e.g. in the lecture notes by Ghys
http://www.math.ethz.ch/~bgabi/ghys%20groups%20acting%20on%20the%20circle.pdf
Section 4.1 has a list of all connected groups acting faithfully and transitively on the circle or the line. They are 
1) $\mathbb{R}$ acting on itself, 
2) the circle acting on itself, 
3) the affine group of the line acting on the line, and 
4) the k-fold cover of $PSL(2,\mathbb{R})$ acting on the circle. 
Any faithful action of a connected Lie Group on the circle is made out of these: if $F$ is the set of fixed points, then on each connected component of the complement of $F$ the action must be conjugate to one of the above.
