Neutron staff type stuff is obviously forbidden because it can’t exist (as neutron star material) without the powerful gravity of a star keeping it together.
What is the absolute physical density limit for an element or substance to exist as stable matter on the earth’s surface?
I’m resisting a very powerful urge to make a political comment here.
As for your question: iridium - 22.65 g/cm3
I should have been more specific. I know what the densest element is, I’m asking what the projected maximum density limits are for a substance that would be able to exist on the surface of the earth without violently self destructing (as neutron star material would do if magically transported to the earth’s surface).
If iridium is the densest element I would assume any other element added to it would decrease its density. So presumedly pure iridium is the densest natrual material. The only way to make a denser material would be to apply external pressure to compress it, which would seem to violate the conditions set in the OP.
I’m guessing here, but I think what the OP is asking is purely theoretical. I also think it is a two part question:
What is the densest material that could withstand a gravitational accelaration of 1 ‘g’ (planet Earth’s surface) and still be able to manitain its molecular (or atomic ?) integrity?
What is the densest material that could be placed on the Earth’s surface before its extreme density would cause it to “bore” itself right down to the Earth’s core?
More or less # 1. I used “earth’s surface” more for the gravity present in that context than for the tensile strength of the crust.
I wasn’t going to answer because I don’t have hard facts (i.e. CITEs 
) to back me up, but I don’t believe the premise underlying the OP. True, you need a stellar mass (at least, I think you do) to create neutronium in the first place, but if you have some already I don’t believe you need a stellar mass to sustain it - whether it’s a pinhead or a star you still have the intense gravity at the surface of the neutronium sustaining it. If I’m incorrect, I fail to see how tiny black holes (suggested in a number of articles I’ve read) could sustain themselves.
So, that leaves the other question - what is the densest material that could sit on the surface of the earth without sinking to the centre? I dunno 
The issue of the sustainability of neutronium (magically transported) to the earth’s surface was explored at some length in this thread
Neutron Star Material Properties on Earth?
and the consensus was that it was utterly impossible for it to exist as neutronium outside of the gravitational influence of a very strong gravitational point source (ie collapsed star).
Sorry to continue this, but:
…says wikipedia (yeah I know, not so authoritative.)
You mean like a politician so dense his head has an event horizon?
That seems like a common sense argument, but from what little I know of chemistry, it’s almost certainly wrong. The density of a material will basically depend on two things
[ol]
[li]Molecular weight[/li][li]Molecular structure[/li][/ol]
Irridium/osmium may be dense for elements at STP, but there’s no gurantee that you couldn’t pack material in even tighter if you had some other atoms in there to change the shape of the molecule and how the bonds fit together.
For example, water is significantly more dense than either elemental Hydrogen or Oxygen.
I’m going to make a half-serious, half-assed observance here:
I would submit to you, that your threshhold of “densest material” would depend on the strength of whatever you’re placing the sample on. If you place a typically solid object on the surface of the water at sea level, it sinks, thus, your experiment is over. If you place a dense enough material on a table, and it breaks the table and goes through the floor, your experiment is over. If you pick up that same material and place it on bare soil, depending on the characteristics of the soil density (think CBR here), your material theoretically might just as well sink into dry, bare dirt.
I throw this in there in that I would personally consider shaping, mass, and use of the material in it’s existence as a “stable material on the earth”.
Tripler
Look! I play hockey. I just threw what I thought was a curve ball!
I guess Tripler is right!
Sounds like strange quark matter has similar density to neutronium, but is stable in smaller quantities.
A neutron star a few miles in diameter maintains its high density by its own mutual self-attraction or gravity. Having one sitting on the surface of the earth wouldn’t change that (though you might debate whether the star was on the earth or the other way 'round).
It wouldn’t take long for the earth to be destroyed as it fell into the star, and that begs the question of how one would get the star there in the first place without having destroyed the earth in the process.
But wouldn’t this count, in terms of the OP?