Today I came across a question (in one of my books) like
Can the compounds $\ce{NOF3}$ and $\ce{POF3}$ both exist? Why?
After some googling I found that both of them do indeed exist. I can understand that $\ce{P}$ can extend its octet due to empty d orbitals. But $\ce{N}$ can't do so. So how can it possibly exist?
Both molecules can be considered the acid (hydrate) trifluoride of the respective acid (phosphoric acid or nitric acid). In the case of phosphoric acid, this is immediately obvious: Just replace the three hydroxide groups with a fluorine atom, each. For the nitrogen compound, you initially need to (formally) add water before you can do the substitution:
$$\ce{O=N+(OH)-O- + H2O -> [(HO)3-N+-O- ] -> F3N+-O-}$$
(Note: this is not a proper chemical reaction, merely an illustration of the thought processes.)
You could also approach this compound from the ammonium cation $\ce{NH4+}$ by replacing three hydrogen atoms with fluorine and the fourth with oxygen. Or you start off with $\ce{NF3}$ (isoelectronic to $\ce{PF3}$ and then oxidise the nitrogen by adding a single oxygen atom much in the same way you would oxidise from $\ce{HNO2}$ to $\ce{HNO3}$.
In fact, for many of the compounds traditionally explained with d-orbital participation for main-group elements, an explanation adhering to the octet rule (no more than eight valence electrons per main-group atom; two for hydrogen/helium) is often closer to the truth. This includes, but is not limited to $\ce{OPF3, SO4^2-}$, sulphonic acids and many more.
Yes, trifluoramine oxide, $\ce{\overset{-}{O}–\overset{+}{N}F3}$, does exist.
The structure written above is the "conventional" structure with sigma bonds indicated between all four directly connected pairs of atoms. It thus appears that there can be only sigma bonds. Yet the length and spectrometric chatacteristics on the nitrogen-oxygen interaction seem to indicate multiple-bond character.
What happens is explained by molecular orbital theory involving only the usual $s$ and $p$ valence orbitals contributing. The $\ce{\overset{-}{O}}$ function described above is a powerful $\pi$ electron donor; its lone pairs with the proper symmetry can actually overlap with the otherwise antibonding components of the $\ce{N–F}$ bonds. This loses some of the $\ce{N–F}$ bonding, but strengthens the $\ce{O–N}$ bonding giving that interaction some $\pi$ bonding character. The latter effect is stronger given the right combination of input atomic orbitals.
We can think of this as a form of resonance hybridization and bond delocalization, in which nitrogen-fluorine sigma bonds are exchanged for nitrogen-oxygen pi bonds:
$\ce{\overset{-}{O}–\overset{+}{N}F3<-> [O=\overset{+}{N}F2]F^-}$
Note the role of the electronegativity of the fluorine, both stabilizing the negative charge imparted by the $\pi$-acceptance and placing more of the antibonding-orbital electron density on the nitrogen where it overlaps the $\pi$-donor oxygen.
This type of bond delocalization may also be rendered with later-period elements, as in the phosphate or sulfate ion where it occurs simultaneously along all four bond axes; the fact that this model also covers $\ce{NOF3}$ is one reason it has displaced the "outer-$d$ orbital" theory.
