What are all the non-split Lie (and topological) group extensions $0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0$? Here, $\mathbb{R}$ and $\mathbb{R}^2$ are regarded as Lie (and topological) groups with respect to the usual addition. One example of a non-split extension is the Heisenberg group $H\_3(\mathbb{R})$ (Please see a post by Alain Valette at <https://mathoverflow.net/questions/63630>). 
Since, every abelian topological extension of $\mathbb{R}^n$ by a locally compact abelian group is trivial, we have that every abelian topological extension of $\mathbb{R}^2$ by $\mathbb{R}$ is trivial. Hence, we need to see only non-abelian extensions.



Lie (and topological) group extensions of $\mathbb{R}^2$ by $\mathbb{R}$
Central extensions 
$$
0 \to \mathbb{R} \to G \to \mathbb{R}^2 \to 0
$$
in which $G$ is a principal $\mathbb{R}$-bundle over $\mathbb{R}^2$ (I suppose you mean that by "topological")
are classified by continuous maps 
$$
f: \mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}
$$
satisfying
$$
f(x,y)f(y,z) = f(x,z).
$$
The abelian ones are those corresponding to maps with $f(x,y) = f(y,x)$. 


This follows from a general theory for topological central extensions described in J.-L. Brylinksi's "Differentiable cohomology of gauge groups" (for the smooth case, but that is not relevant) combined with the fact that every principal bundle over $\mathbb{R}^2$ is trivializable.


EDIT: From a map $f$, you get the extension $G$ *explicitly* as the topological space $G = \mathbb{R} \times \mathbb{R}^2$ with the multiplication given by
$$
(a\_1,x\_1)(a\_2,x\_2) = (a\_1 + a\_2 + f(x\_1,x\_1^{-1}),x\_1 + x\_2).
$$


