Why, for example, does nitrogen not do this, forming, say a polymer (or smaller molecule) of formula (NH)_n, ditto phosphorus? I am an experienced Chemistry teacher but I have never seen an explanation for this.

Isaac Asimov wrote a very good article that gives a comprehensive answer to your question and several that are closely related. It is called "The One and Only" and appears in his essay collection The Tragedy of the Moon (now out of print). This answer is largely based on Asimov's; he seems to have anticipated every point I wanted to use!

It all comes down to thermodynamics.

Specifically, for an element to link to itself in long chains, you need three things:

1. The ability to form more than one bond per atom. This excludes hydrogen and the halogens, for example. We will also exclude the metals, since they don't form localized covalent bonds to themselves and so cannot form chains of atoms.
2. A preference for single bonds over multiple bonds. That is, two single bonds should be more energetically stable than one double bond, and three single bonds than one triple bond.
3. The oxide must not be too much more stable than the pure element. Oxygen is the third most common atom in the universe, and an overwhelming tendency to bond to oxygen will mean that an element will never be found except as the oxide.
4. Finally, the bond to hydrogen must be reasonably strong, compared to the hydrogen-oxygen bond. Hydrogen is the most abundant atom in the universe and (as you indicated in your question) is commonly used to "cap off" chains of atoms. But many compounds of hydrogen are spontaneously flammable in the presence of oxygen!

Single vs. Multiple Bonds

Information in this table is taken from M.A. Fox and J.K. Whitesell, Organic Chemistry, 2^nd Ed., Jones & Bartlett, 1997. Values are averages and will differ somewhat depending on the source of data.

Notice that it is favorable for carbon to form singly-bonded chains. For example, two C-C bonds are worth more than one C=C bond: 2 × 83 = 166 > 146.

But nitrogen and oxygen, though formally capable of forming extensive chains of atoms, cannot do so because of the weakness of N-N and O-O single bonds. Quite simply, three N-N bonds are so much weaker than one NºN bond (3 × 39 = 117 << 226) that polyazines (NR)_n are explosive! The same is true for peroxides, compounds containing O-O single bonds.

Of course, once you get below the first row multiple bonding becomes much less energetically favorable. Silicon, phosphorus and sulfur can all be found in long, singly-bonded chains or network compounds. The most stable forms of elemental silicon, phosphorus and sulfur (S₈) contain polymeric chains or rings of atoms bonded to each other by single bonds. This is also true for boron, arsenic, selenium and tellurium.

Bonds to Oxygen

Boron-oxygen, silicon-oxygen and phosphorus-oxygen bonds are so much stronger than their homonuclear bonds that they are never found in nature as pure elements. Elemental carbon and sulfur, however, can exist for a very long time in the presence of air.

Arsenic, selenium and tellurium, the only other non-metals capable of forming chains, are thermodynamically unstable in the presence of oxygen. All three exist as pure elements in polymeric chains or rings of several atoms--but not in nature!

Bonds to Hydrogen

Hydrides and halides of main-group elements (other than the fluorides) are typically quite unstable in the presence of oxygen or water. This is much less true for carbon; while boron and silicon hydrides spontaneously combust in the presence of oxygen, and phosphine (PH₃) is highly reactive, the half-life of methane (CH₄) and other saturated hydrocarbons in air is a couple of centuries. Ammonia (NH₃) has a stability similar to that of methane.

However, polymers of the form (EH_x)_n are highly unstable in the presence of oxygen for every element other than carbon, normally the more so, the longer the chains. For nitrogen polymers it's because multiplying N-N single bonds increases reactivity, but for other elements such as silicon, phosphorus and arsenic the instability is primarily the result of weak element-hydrogen bonds.

Boron is a special case. Hydrogen atoms in boron hydrides (BH_x)_n are typically shared between two or three boron atoms, and since oxygen has so many more electrons available, the reaction of boron hydrides with oxygen is powerful enough to be used for rocket fuel!

Most of the information in the final parts of this answer was taken from N.N. Greenwood and A. Earnshaw, Chemistry of the Elements, Pergamon, 1984.