Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal such group, for all possible trivialisations?
Another way of phrasing the question: am I correct in thinking that there are potentially many $G$-bundles which are the same as fibre bundles?
Yes, the structure group is not unique. For example, a vector bundle $E$ has, by definition, a structure group $GL(n)$. Some additional structures on $E$ are equivalent to reductions of the structure group to a subgroup of $GL(n)$ and (even if they exist) you may not want to specify these additional structures and thus may not care about getting a smaller structure group.
For example $E$ will always admit a metric (at least if the base is nice like a manifold) and specifying a metric is equivalent to giving a reduction of the structure group to $O(n) < GL(n)$ (given a metric you consider only orthonormal local frames, which shows that the transition functions take values in $O(n)$ and conversely given a reduction to $O(n)$ you take an $O(n)$ trivialization of the bundle and define a metric by making those frames orthonormal). If you don't care about using a metric then you may not want to reduce the structure group (which requires a choice and so is not natural).
Another example is if the bundle is trivializable then its structure group is the trivial group. Again, there may not be a natural trivialization so you may not want to think of the structure group as trivial.