Is there a topologizable group admitting only Raikov-complete group topologies? Definition. A group $G$ is called complete (resp. non-topologizable) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.
Question 1. Does there exist a complete topologizable group?
In particular:
Question 2. Is the group $SO(3,\mathbb R)$ complete?
Question 3. Is the group $Sym(\mathbb N)$ complete?
A simple Baire category argument shows that each complete topologizable group is uncountable. 
Remark. There are many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.

In particular, $Sym(\mathbb N)$ is such a group.
 A: A simple way to come up with non-complete topologies on $\text{SO}(3,\mathbb{R})$ is by embedding
$$ \text{SO}(3,\mathbb{R}) \to
\text{SL}(3,\mathbb{R})\to
\text{SL}(3,\mathbb{C})$$
and then using a discontinuous automorphism of 
$\mathbb{C}$ to produce a discontinuous automorphism of 
$\text{SL}(3,\mathbb{C})$.
Knowing all the (not so many) closed subgroups of $\text{SL}(3,\mathbb{C})$, it is easy to deduce that the image of $\text{SO}(3,\mathbb{R})$ under such an automorphism is not closed, hence the induced topology not complete.
Another nice way to finish the argument is by making an identification of $\mathbb{C}$ with another algebraically closed field of the same cardinality, carrying a different topological structure. For this one can use the "$p$-adic complex field", $\mathbb{C}_p$.
One gets the embedding
$$ \text{SO}(3,\mathbb{R}) \to
\text{SL}(3,\mathbb{R})\to
\text{SL}(3,\mathbb{C})
\to
\text{SL}(3,\mathbb{C}_p).$$
Finally, to cope with Andreas' remark, note that
$\text{SL}(3,\mathbb{C}_p)$ has a countable permutation action. Indeed, this Polish group has an open subgroup, namely
$\text{SL}(3,\mathcal{O})$, where $\mathcal{O}<\mathbb{C}_p$ is the ring of integers.
