Are there languages that are not in RE nor CO-RE? I am familiar with the theorem which states that some languages are not in the RE (Recursively Enumerable) class of languages, but that can mean either that they are all in CO-RE (or rather, the part of it that doesn't intersect with RE), or that they are partly in CO-RE and partly somewhere else. 

Are there languages about which nothing can be decided, not even what words are not in them?
Yes, there are some. There is actually an infinite hierarchy of languages which are less and less decidable, namely the Arithmetical hierarchy. Recursively Enumerable languages and their complements are at its level 1. An example of language which isn't RE or coRE is the set of Turing machines computing total functions.