Atomic nucleus consisting of only neutrons? Is it true that the nucleus of all atoms (including radioactive isotopes) contain at least one proton? Is there an atomic nucleus consisting entirely of neutrons? (Let's exclude neutron stars for the moment.) If so, how does one name them? (since Periodic Table starts from atomic number 1, not 0.)
 A: Well, actually there exists a nucleus which contains no protons. It has the atomic number 0, the mass number 1, and consists of one neutron, zero protons and zero electrons. It is called neutron. It is an unstable nucleus which decays via beta decay.
If you think that calling the neutron a nucleus is not proper, then think of the following: The hydrogen nucleus is just a proton. And chemists have no problem to talk about $\mathrm{H}^+$ ions, which are also nothing but protons, without any electrons around them.
According to the Wikipedia page DarioP linked to, also a di-neutron ($Z=0$, $A=2$) has been observed, which is extremely unstable. While the decay channel is not stated there, I guess the two neutrons just separate from each other; whether you call that neutron emission of spontaneous fission is a question of semantics. I guess in principle there would be also the possibility of beta decay to deuterium, but I'd not expect that to happen in observable rates.
Higher isotopes have not been observed, and are not to expect from theory.
A: Such thing can exist. However, we call atoms the elementary particles of what we consider "matter", which have some physical and chemical properties.
Chemical properties are most important here, and they are defined by the atom's electron configuration — actually, everything chemical that happens in the entire world happens due to atoms interacting through their outer electron shells. "Nucleus" that consists only of neutrons has no charge and cannot hold any electrons, so it is really just a bunch of neutrons in space. Probably this is why people don't call a single neutron  "nucleus" (which would imply that it is the inner part of "something", but there is no something for such nucleus). Not so say that free neutron decay (AFAIR, the half-life is about 10 minutes).
A: To name it, you just do it and extend the periodic table to start from zero.  (the periodic table is a human invention and we can modify it, if we want.) 
Let's call it Zeronium (we can give it atomic symbol Z, it's available!).  Having no electrons, it would be chemically inert, so it belongs above He on the periodic table. 
We know that Z-1 is unstable with a half-life of about 10 minutes, which is glacially slow by particle physics standards. (it's also long enough that we can imagine studying its chemical properties)  
We also know that Z-infinity (neutron star) is stable.  For all other isotopes of Zeronium, I have no idea if it's stable or not.  It requires understanding the https://en.wikipedia.org/wiki/Nuclear_shell_model 
The chemical properties would be interesting.  At room temperature, presumably Z-1 is a gas.  But does it freeze?  What is the boiling point?  What is to stop the nuclei of very cold condensed Z-1 gas from coalescing into heavier Z-n isotopes?
If there are any stable isotopes of Z, does that explain dark matter?  
My hunch is that there is some good reason why Z-n is unstable, If there were stable forms, we would have heard about them.
But there are serious practical problems with studying Z-1.  To start, it's neutral and has no electrons, so it doesn't interact chemically or with light. High energy neutrons can be detected indirectly, but ones with only thermal energy are very hard to even see, much less control. Then there is the problem of neutron capture, where ordinary atoms absorb free neutrons and become a heavier isotope.
Arguably, given the neutron capture problem, free thermal neutrons don't fit the definition of an atom very well, which makes a good philosophical case for leaving them off the periodic table.  (If atoms are nouns, neutrons are more like verbs, which go around modifying nouns) 
Finally, there is a very practical reason for leaving it off. When teaching kids basic chemistry, the periodic table makes sense and organizes our knowledge and every-day experience.  I can have a good conversation with a clever 6 year old about the first 5 rows of the periodic table.  Basic concepts like elements, compounds and chemical reactions become clear.  
But I'd lose that kid if I introduced radioactivity.  Why?  Because permanence is inherent to the definition of an element.  The intrinsic properties of an element are preserved in any chemical reaction. You gotta teach them basic chemistry before springing the bad news that okay, some elements spontaneously turns into other ones, and it's all the neutrons fault! 
So while we could put it at the top of the periodic table, we don't.  It makes more sense as a footnote, in places like this, than on the wall in every chemistry class.       
