What makes ammonium nitrate soluble?
Here is some data on other well-soluble salts:



Comparing the values, both the anion and the cation contribute to magnitude of enthalpy and entropy. The larger cations seem to contribute to a larger entropy of dissolution, as do the larger anions. The entropy is concentration-dependent, so I assume the data is for a standard state of about 1 mol/L dissolved salt.

There is a nice discussion about the dissolution of ammonium chloride here. The enthalpy of dissolution (14.4 kJ/mol) and the entropy of dissolution (75.3 J/(K mol), source) are close to that of KCl, so there is nothing super-special about the ammonium ion.


  [...] since their ΔH values are usually much larger than the TΔS values they would have, especially at room temperature.


At the beginning of the dissolution process, the entropy dominates. At the solubility limit, entropy and enthalpy contributions are equal and add up to zero.


  What makes ammonium nitrate soluble?


I did not find a data table that includes entropy and enthalpy of dissolution for ammonium nitrate, but you could easily calculate these from entropy and enthalpy of formation of the reactants and products. On second view, I am having trouble finding the data for aqueous nitrate, but this textbook has some values. There is a report that nitrate comes in two forms in aqueous solution, perhaps complicating things.

Some people seem to think that the dissolution of nitrates is special in terms of enthalpy (compare to e.g. carbonate) and others argue that the entropic contribution is special. It would be easier to answer the question, for example, "what makes ammonium nitrate more soluble than ammonium carbonate".