Why does N(OH)3 not exist?
Although the acid and its salts are unknown, orthonitrite esters have been found, and from a natural source at that [1].  The orthonitrite function, which is trivalent rather than monovalent as nitrate would be, is incorporated into a macrocycle that hinders its decomposition.

The picture below is from Ref. 1.

Reference:

1.
Leanne Murray,* Tang K. Lim,* Graeme CurrieB and Robert J., "Aplidites (A-G): Macrocyclic Orthonitrites from 
an Australian Tunicate, Aplidium sp.", Aust. J. Chem., 1995, 48, 1253-1266.Link