For some odd reason, I was sure there was an independent kingdom but clearly not. I think I've played too much Ck2 to remember my real history now. Thanks for that :)
Well, like I said, there was the High Kingship, but opinion seems to be divided on whether it represented a genuine political union, a confederation, a series of occasional and temporary alliances, a courtesy title for a "first among equals", a claim of supremacy not necessarily accepted by other Irish kings, or all of the above at different times. And at least some of its incumbents, especially the earlier ones, were entirely fictional.