5 corrected own blunder in comments to another answer; deleted 1 characters in body

You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint open subset. So the additive group $\mathbb R^N$ becomes a subgroup of $Diff(S^1)$ by flowing along $N$ commuting vector fields supported in $N$ disjoint arcs.

(added later) A similar cheat: let $a_1,\dots ,a_N$ be linearly independent functions of one variable. Then $a_1(x)\frac{\partial}{\partial y},\dots ,a_N(x)\frac{\partial}{\partial y}$ are independent commuting vector fields in the $x,y$ plane. Modify this example to make it compactly supported if you like.

(added still later) My proposed extension of the first cheat to a semisimple group (comment thread of Torsten's answer) is doomed: Choose a point on the circle and choose an element of SL_2(R) that fixes this point and acts on the tangent space there with eigenvalue c>1. Lifting the group element to the universal covering group in the right way, you get an element g of the latter group that fixes all the points above the given point in the universal covering space of the circle, in each case with eigenvalue c. But now if this line with this action could be embedded in a longer line with trivial action outside then there would be a sequence of fixed points of g with eigenvalue c converging to a fixed point of g with eigenvalue 1, contradiction.

4 removed false statement

You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint open subset. So the additive group $\mathbb R^N$ becomes a subgroup of $Diff(S^1)$ by flowing along $N$ commuting vector fields supported in $N$ disjoint arcs.

(added later) A similar cheat: let $a_1,\dots ,a_N$ be linearly independent functions of one variable. Then $a_1(x)\frac{\partial}{\partial y},\dots ,a_N(x)\frac{\partial}{\partial y}$ are independent commuting vector fields in the $x,y$ plane. Modify this example to make it compactly supported if you like.

(added still later) More generally, let $g$ be a $d$-dimensional Lie algebra of vector fields in the coordinates $x_1,\dots ,x_{n-1}$ and let $f$ be an $N$-dimensional vector space of functions of $x_n$; then $g\otimes f$ becomes an $Nd$-dimensional Lie algebra of vector fields in the coordinates $x_1,\dots ,x_n$

3 added third paragraph

You can make big Lie groups act effectively on small manifolds by cheating: make the group a product of groups, with each factor acting by compactly supported diffeomorphisms on a different disjoint open subset. So the additive group $\mathbb R^N$ becomes a subgroup of $Diff(S^1)$ by flowing along $N$ commuting vector fields supported in $N$ disjoint arcs.

(added later) A similar cheat: let $a_1,\dots ,a_N$ be linearly independent functions of one variable. Then $a_1(x)\frac{\partial}{\partial y},\dots ,a_N(x)\frac{\partial}{\partial y}$ are independent commuting vector fields in the $x,y$ plane. Modify this example to make it compactly supported if you like.

(added still later) More generally, let $g$ be a $d$-dimensional Lie algebra of vector fields in the coordinates $x_1,\dots ,x_{n-1}$ and let $f$ be an $N$-dimensional vector space of functions of $x_n$; then $g\otimes f$ becomes an $Nd$-dimensional Lie algebra of vector fields in the coordinates $x_1,\dots ,x_n$

2 Added second paragraph
1