Apparently, the closest thing I've found would be normal number http://mathworld.wolfram.com/NormalNumber.html
But requiring that every finite words occurs is weaker than this property. So I'm wondering if there are any study on this topic.
My original goal is to find a criterion for a Büchi automaton not to recognize some infinite word like this. There's been a post here Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular , but there were no references to such a class of words.
Maybe all this is obvious to specialists, but I couldn't find anything with our universal friend google :)