$\omega$-topos theory? I've been reading through Lurie's book on higher topos theory, where he develops the theory of $(\infty,1)$-toposes, which leads me to the following question: Is there any sort of higher topos theory on the more general $\omega$-categories, where we don't require all higher morphisms to be invertible?
 A: The short answer is no.  Even 2-toposes are poorly understood -- we don't know what the right definition is.  For higher dimensions, including $\infty$, we definitely don't have the answers.
Just as the primordial example of a (1-)topos is $\mathbf{Set}$ (the 1-category of sets and functions), the primordial example of a 2-topos should be $\mathbf{Cat}$ (the 2-category of categories, functors and natural transformations).  
Mark Weber has done some work on 2-toposes, building on earlier ideas of Ross Street.  But I think Mark is quite open about the tentative nature of this so far.  
There was a good discussion of the current state of 2-toposes (and more generally n-toposes) at the $n$-Category Café last year: 
A: One should keep in mind that Jacob Lurie's book "only" (if I may use this word) discusses the (oo,1)-version of Grothendieck toposes/category of sheaves: the (oo,1)-toposes in Jacob Lurie's book are (oo,1)-categories of (oo,1)-sheaves/of oo-stacks.
This is less general than the "elementary (oo,1)-toposes" that one would eventually want to see, but it already goes a long way -- and it is more accessible.
Similarly, while a general theory of n-toposes for higher n is largely missing, there is a bit more known about (oo,n)-sheaves, i.e. of oo-stacks which are presheaves with values not just in oo-groupoids but in (oo,n)-categories. 
For instance Ross Street once proposed a notion  of descent for strict-omega-category-valued presheaves. Using a result by Verity this may be regarded as presenting descent for strict oo-groupoid valued presheaves, but I'd expect that with the required care exercised it goes further than that (and this seems to be what Street had in mind, though I can't tell that for sure). 
But a more developed general theory for descent of (oo,n)-category valued presheaves is developed notably in Hirschowitz-Simpson's Descente pour les n-champs. This yields at least part of a theory of Grothendieck-style n-toposes.
A: Mike Shulman has been thinking about n-toposes in general and 2-toposes in particular.
