I have been getting into infinitary logic recently, and stumbled upon $\mathcal{L}_{\omega_{1},\omega}$ and $\mathcal{L}_{\omega_{1},\omega_{1}}$. In particular I've been reading Dickman's "Large Infinitary languages" and Marker's infinitary logic book. I was particularly struck by the undefinability of well-ordering - a "fragment" of compactness - being sometimes used in place of compactness when working with $\mathcal{L}_{\omega_{1},\omega}$.
In that sense, I was wondering whether there are any known failures of expressiveness of $\mathcal{L}_{\omega_{1},\omega_{1}}$ which relate to compactness properties, or some analogous. I have a hunch that probably no such thing should exist, given how powerful the logic is, but I'd be interested to hear anything about this!