The stability of TEMPO and hence is formation is thought to result from resonance involving the nitrogen lone pair as well as hyperconjugation from the numerous methyl groups. See also the Wikipedia article on 3 center 2 electron bond.
The resonance involves oxoammonium contributors as those postulates in oxidation catalyse by TEMPO and carbanions, hence invoking sacrificial hyperconjugation.

As stressed in other answers the steric hindrance of the methyl groups also plays a major role "caging" the radical center and leading to longer than usual bonds when TEMPO reacts.
Consider that coupling two TEMPO would lead to a peroxide. Peroxides are, with few exceptions, unstable and reactive.
The above effects makes a hypothetical $\ce{O-O}$ bond in TEMPO even less stable than that one in other peroxides, where it is already characterised by energy of about $\pu{50 Kcal/mol}$, i.e. less than the already unstable $\ce{O-H}$ bond in the same molecule and generally speaking less than a half of $\ce{C-C}$, $\ce{C-H}$ and $\ce{C-O}$.