You could write a story about the war between the Vulcans and the Romulans, for one. That's never been explored in any depth. Heck, that could be an entire series by itself, with almost no risk of significantly violating the canon.
While I fully agree, this series wouldn't involve a single human. I don't think this would really work, since it's harder to relate to some foreign species, especially as alien as Vulcans and Romulans (to the general public, anyways). Note how a large majority of the main characters in every single Star Trek series and film were human, even though plenty of alternatives were available.
I personally would love to see Klingon everyday life before the warrior caste took over the whole social structure.