One Product Planet: Averted, in its strict form. The realities of interstellar economics, logistics, and costs of transportation mean that it’s almost always more practical to maintain a decent-sized agricultural and manufacturing base at home, rather than import all your food and goods, for anything but the smallest of outposts.
Played straight in a loose form, in which certain worlds are known for certain of their (mostly unique) products, for example:
Big Dumb Object: Within the Empire, the partial Dyson sphere at Corícal Ailek (which exports thought) and the Dyson bubble at Esilmúr (which exports antimatter and other forms of stored energy in unwholesomely large quantities) would qualify.
Capital: For the Empire, that’s Eliéra, the throneworld, which does indeed export governance – to such extent as the Imperial governance is all that centralized, and indeed, can be bothered to govern. For the Worlds as a whole, that would be Conclave (Imperial Core), where the Conclave of Galactic Polities sits and attempts to bring some order to the chaos, with all the associated politicking, corruption, intrigue, and scandal you might expect.
Exotic: A number of these, from the shell-world of Thalíär (Principalities) – mostly exotic from the point of view of the tourism industry – to the blue-white giant in the Ringstars and the black hole out in the Last Darkness constellation. Also, certain exotic matter products are primarily manufactured near the high-energy environment that best supports them, and so have major factories out by Esilmúr, also.
Factory: Qechra (Imperial Core), a corporate conlegial colony world completely overtaken by autoindustrialism, with a manufacturing capacity of holy crap how much!?. It’s more of a showpiece than anything else, and secondarily a place to manufacture ridiculously large items, but it also serves the valuable purpose of being a worrisome sleeping giant.
Farm: Yes, in the sense that there are more than a few worlds that take pride in exporting their local specialist products, from specialist flowers to unique local booze. No, in the sense that just about every world, or at least system, can manage to feed itself locally, and there are no worlds absolutely dependent on their imports of agricultural products, or mighty grain-ships ploughing the spacelanes.
Gates: The closest you get to this are the systems in any given constellation which house the long-range gates to other constellations, and thus are about as close as anything gets to being bottlenecks. (For the Imperial Core, these are Almëa, Meryn, Ocella, Sy, and Vervian Systems).
Military: It doesn’t export military forces – if anything, it imports them – but the Palaxias (Imperial Core) system is essentially given over to the Imperial Navy and its Prime Base, which also houses a large amount of the rest of the Imperial Military Service by default. Also, to a lesser extent, the six systems out in the fringes where the IN keeps the mobile naval bases for its sextant fleets.
Mines: See once again the Imperial energy production facility at Esilmúr; aside from such rare and specialized facilities as it, though, resource gathering tends to be distributed all over the place.
Science: The corporate conlegial research colony at Wynérias (Imperial Core) is notorious for its pursuit of unrestricted research FOR SCIENCE!, as is – even more so – the private conlegial colony at Resplendent Exponential Vector, but they’re hardly the only place where Science happens. Or even where FOR SCIENCE! happens.
Service/Cultural Center: Most notably, Seranth (Imperial Core) is the largest and most prosperous tradeworld the Empire, or even the Worlds, have to offer. It’s by no means, under the general principle I mentioned above, an entire planet of Wall Street, but the Seranth Exchanges do dominate the local economy, and the floating cities of Seranth probably are All Manhattan, All The Time. It’s a very dominant commercial center, and only just eclipsed as a cultural center by Delphys (Imperial Core) for entertainment and art, and Viëlle (Imperial Core) for media in general.
Underworlds: Nepscia (Galith Waste) is infamous throughout the Worlds for its red market. Litash (Dark Sea) was even more infamous for both that and acting as a major pirate center, before it got strangelet-bombed out of existence.