What if I shot protons and electrons at a neutron star Could I create a massive element simply by putting protons into a neutron star? And then forcing electrons to orbit it?
 A: It turns out that through the "weak nuclear interaction" a free neutron sitting in the middle of space will decay into a proton, an electron, and an electron antineutrino. This is actually pretty fast, with a half-life of about ten minutes: so if you put a hundred neutrons into a diffuse neutron gas in a sealed box you'll find about 50 Hydrogen atoms after about 10 minutes, about 75 after about 20 minutes, or about 83 after 30 minutes.
The same mechanism at play in a neutron star will act in reverse, because the energies work the other direction: a proton and electron will spontaneously combine to form a neutron and an electron neutrino; or a proton by itself may spontaneously form a neutron plus positron plus electron neutrino.
A: Following the comment that I misunderstood the question there are still a few reasons why the answer is "no."
Atoms are fundamentally a chemist's abstraction that enable us to talk about molecular structures and so forth. From a really abstract theoretical physics perspective, there's just up- and down-quarks and electrons in conventional matter, and no reason to invoke "atoms" per se at all. (This is the same perspective that you might take if you wanted to argue that a candle flame "isn't real:" it is an abstraction of an active process which happens to look, from a certain perspective, relatively still and timeless, but strictly speaking the flame is not an object in its own right and the real objects are the constituents of the process.)
Therefore, the value in calling this thing an "atom" is "does it do the atomey things that atoms do that we use this abstraction for?" and the answer is basically "no." For example even if you could isolate the neutronium from the core of a neutron star, shove a proton into it and make an electron orbit it, chances are it would not have a hydrogen spectrum the way hydrogen, deuterium, and tritium do, because your electron orbit would be much too classical. I suppose you could let the electron drop to the ground state but then you risk beta-decay, as mentioned above. Since the gravitational force cannot be screened or otherwise balanced, you do not have things like the H2 molecule, either -- at least not without making the nuclei orbit each other as well.
Even worse, the neutronium at the core of a neutron star is not necessarily the whole picture of the neutron star! It is generally believed that neutron stars have atoms at the surface, where the pressures are lower: probably some iron, maybe some hydrogen and/or helium. It would seem weird to describe, say, a big sphere of iron with a single excess proton charge as an "atom" and similarly it's weird to describe a charged neutron star as one. But if you take the neutronium out of the neutron star it presumably explodes immediately.
Even worse, this definition of "an atom is some electric charges orbiting some positive charges" probably fails to classify the Hydrogen family as atoms (H+ has no electrons, H has an electron but it technically doesn't have any orbital angular momentum, H2 probably falls into an ambiguous case where we can't say that the electrons are necessarily orbiting either of the two rather than, say, figure-8 orbits or whatever, because quantum mechanics won't give us discrete orbits.)
