Class of classes not class? It's well known that $X=\{x: x\in Set, x\notin x \}$ is not a set, so set of sets may not be set, and I have been told this $X$ is a class.
But the same procedure applied to class, $Y=\{y: y\in Class,y\notin y \}$, then $Y$, a class of classes is not a class, so is there some structure transcends class, maybe called upperclass? It goes on and on and hence we need upper upper upper... upper class?
It's nature that we want to decide wether a thing is in some collection, so to avoid logic flaws of collection of things, we need infinite many structures?
I have wondered this for a long time but have no time diving into set theory, can someone explain a little bit without heavy use of set theory which is not familiar by most math students?
 A: In category theory collections of classes are called conglomerates.  Likewise there are structures larger then conglomerates creating an unending upwards chain.  However there has been no real study of collections larger then conglomerates so no one has given them a name.
A: Lets look at it simply typed. Some classes, that are not proper classes are sets, but lets put aside sets for the moment. Lets assume you have a type for individuals i and a type for properties o, then the classes have the type:
classes : i -> o

Then you have:
classes of classes : (i -> o) -> o

classes of classes of classes : ((i -> o) -> o) -> o

Etc..

In Gödels incompletness paper (*), this was called a type level, and he used numerals 1, 2, 3, ... to indicate type levels. But what happens to the sets. Lets say, the non-proper classes are the sets, so there is a mapping:
 s : i -> (i -> o)

The above mapping maps sets to their non-proper classes. But sets will not go up some levels. Although there is a power set, this is only a mapping p : i -> i, so that with this viewpoint we stay on level 1 with our sets. :-( :-) 
(*)
On Page 176 Gödel uses the German terms:
Klassen von Individuen
Klassen von Klassen von Individuen 
