Elements which do not form oxides
The first thing that came to mind is oxygen itself, as the term "oxide" suggests one other element in its chemical formula. Merriam-Webster suggests a more strict version: "a binary compound of oxygen with a more electropositive element or group". So technically pure oxygen is not an oxide.

Putting oxygen aside, helium $\ce{He}$, neon $\ce{Ne}$ and argon $\ce{Ar}$ are probably the only three conventional elements that come to mind are not yet proven to form oxides. I find it rather tricky to back up this statement properly as the lack of knowledge about an object doesn't prove its non-existence; ideally one might want to do a stability simulation at extreme pressures and temperatures to prove the existence of $\ce{(He|Ne|Ar)O_x}$ or lack thereof. There are algorithms like USPEX that are capable of that – by the way, with the help of USPEX a family of krypton oxides has been recently predicted.