So, “quasi” can be a useful prefix! (Just ribbin’ ya, Minh).
With excellent (continuing) research into the structure of New York City (which is, after all, an OSM-consensus-uttered admin_level=5, a true rarity in the USA), I can see community districts in NYC being higher-numbered admin_level=* tagged where the boundaries are the correct, appropriate closed polygons. We may find we use even more than one admin_level value (>5) inside of NYC’s many possible or actual sub-structures of internal governance. It is fascinating to see these concepts emerging and making their way into good OSM discussion (like this, and the Slack thread Minh notes, and many others…) as well as having the potential to enter OSM itself as actual (>5) admin_level data, should we reach consensus upon a sensible method to do that. Blue skies ahead, friends; keep talking.
To answer Elliott’s question about admin_level=10, you can see by our (deeply incomplete, almost comically complex thanks to the wisdom of the 10th Amendment, yet a bedrock of good, basic structure) United States admin level - OpenStreetMap Wiki that value 10 usually corresponds to “Neighborhood,” (without the British “u” spelling, as this is an American concept) where this is an “organized” (real, rather than imagined or “amorphous”) entity that passes ordinances, rules or otherwise “governs” highly local aspects distinct from other such entities in the same (city, town, often at admin_level=8). The USA does have “neighborhoods” (call them “small n”?) which truly ARE “amorphous” but don’t rise to the bar of what we call a 10, because they aren’t really an administrative boundary. They are something similar geographically, but we don’t tag them with admin_level (of any value) if they are not really administrative, but rather more geographic, and even amorphous at that.