What object in the universe is known as having the greatest angular momentum?
Hmmm. I'm not sure that has ever been tested in detail. If you mean singular object, and exclude collections of objects like galaxies and galaxy clusters, then my bet would be the supermassive black holes at the centers of galaxies have the most. They use this angular momentum to drive massive jets of highly relativistic particles (traveling very near the speed of light), jets that can be as long or longer than the galaxy itself. I'm not sure which one has the strongest jet, but then the problem is that the strength of the jet isn't only a function of the black hole's angular momentum, but also the amount of matter falling into it. So quiet supermassive black holes may well have very large angular momenta, but without matter falling into them it's hard to tell.
I would vote neutron stars. I can see how a black hole can have a rotational axis, but, not a surface. Matter falling into a black hole could acquire significant angular momentum, but, not nearly as much as the surface of a neutron star, imo.
Angular momentum is one of the few properties a black hole does have, though: no hair theorems show that a black hole can only be distinguished by mass, charge, and angular momentum.
And while neutron stars have very high rotational speeds, they really don't have more angular momentum than the stars they are produced from.
How about for an object defined as being gravitationally bound?
Including galaxies and galaxy clusters? Then the winner would almost certainly just be the largest galaxy cluster.
Any one in particular?
Well, I'm just not sure what the most massive galaxy cluster that we've so far detected is. The Coma cluster is pretty impressive, though, coming in at over 1000 so far identified galaxies:
It was observations of this cluster, actually, that first led to the proposal of dark matter by Fritz Zwicky.
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