In their paper titled "The Classification of Hypersmooth Borel Equivalence Relations" Alexander Kechris and Alain Louveau quote the following (Theorem 5.2 in the article) as "Harrington, unpublished":
There is a nonempty class $U$ of Borel equivalence relations which is unbounded, and every $F \in U$ is induced by a Borel action of a Polish group (so in particular is idealistic).
I have been unable to find any proof of that, help would be greatly appreciated.
The only idea (which probably leads nowhere anyway) I had to get higher and higher orbit equivalence relations was to apply the FS-jump to some orbit equivalence relations, since those again can be shown to be induced by Polish group actions. But that's not sufficient for the theorem by any means, so I'm not sure it makes sense to mention this here.