TITLE: Ultrainfinitism, or a step beyond the transfinite
QUESTION [6 upvotes]: Cantor has, in the immortal words of D. Hilbert, given all of  us a paradise (or perhaps, I would  rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists goes on forever, into higher and higher ethereal realms. In his theological mind,  Cantor thought of these dots as an eternal  ladder, which approaches (without ever  reaching it) the Absolute Infinite, later re-christened as $V$, the Universe of Sets, by Set Theory adepts.
Those same adepts have enriched Cantor's paradise with a great bestiary of enormous cardinals, inaccessibles, Mahlo, Vopenka, Woodin cardinals, etc. Big fellows, no doubt. Yet... In comparison with the size of $V$ they are puny, nil in fact, no more no less as Graham number, or Friedman's TREE(3) stand in comparison to (for finitists) almighty $\omega_0$.  
Now, let us be brave and say: what about breaking through into the trans-transfinite?
What about , for instance, starting from $V$ itself and state that its size is some hyperinfinte number, say $\aleph_{0,1}$ ?
(SIDE NOTE ON NOTATION: The standard aleph series would  now be $\aleph_{0,0}$ , $\aleph_{1,0}$, .... The second subindex controls the degree of hyperfiniteness, much like degrees of unsolvability. I could have put it on top, but then it would cause troubles with cardinal exponentiations ).
Wait, I hear you say loud and clear. Are you crazy?
Don't you know that there is NO SET $X$ such that $X=V$?
Don't you know that there is no max ordinal? 
Yes, ladies and gentlemen, I do know it. But I do reply: and so what? The objection is exactly the same as the one of the finitists vis-a'-vis $\omega$. Someone has broken through the finite, so why not the transfinite? There is no set, but who said that it must be a set? 
In fact, start with a pairs of transitive countable  models of ZFC, $M_0$ and $M_1$, with $M_0\leq M_1$, of different tallness (the ordinal height of the first being strictly smaller than the height of the second). From the point of view of $M_0$, IT is the full universe of sets, and the ideal ordinals of $M_1$ some unimaginable higher level of infinity. Of course, say you, $M_0$ does not see $M_1$. 
True, but we do. And -I think- nothing prevents us from formalizing their reciprocal relation as some new theory of sets (the elements of $M_0$) and classes (the elements of $M_1$). Note that here all sets are classes, but not viceversa. 
Also, being  more reckless, we could generalize the above by stipulating an entire chain of ascending hyper-infinities, and perhaps enrich ZFC with an axiom that says that for each model there is a cofinal (in V) ascending chain of taller models, the Cofinal Tallness Axiom....
OK, now the question(s): 

(set-theory) has anything like the above be attempted? 
(algebra) can we create a system of "numbers" which strictly contains cardinals plus other numbers strictly greater than them? And if yes, what is their arithmetics? 

NOTE: by 2 I mean: axiomatize directly the class CARDINALS. Then find a new class of numbers, say HYPERCARDINALS, which contains CARDINALS as an initial segment, and moreover such that the numbers in HYPERCARDINALS - CARDINALS has some arithmetical property that ordinary cardinals, no matter how large, have not (this will rule out simply having copies of cardinals appended after one another). 

(philosophy) is there any speculation as to a radically NEW notion of infinity, which makes all large cardinals small? 

NOTE: this is of course connected to 2 above, but would interpret the new arithmetical/algebraic  characteristics of the hyper-cardinals as speaking of new properties of hyper-infinite classes. Essentially this interpretation would unravel new conceptualizations of the informal notion "being infinite" . Of course, the challenge here is to steer away from blatant inconsistencies, such as the ones discovered in the early history of Set Theory, and which were eliminated in the formalized ZF approach.  
Any reference, thought, criticism, and what not is most welcome.

REPLY [4 votes]: I would go so far as to say one has to not think of things larger than the collection of classes (in so far as such a thing is even defined) not as just 'bigger sets' but something else. I posit that that something else is ... categories! In particular, not just categories in the fairly vanilla sense as a class of objects (which may be a set) and a set of arrows (or perhaps a class) between any two pairs of objects, but using the first order definition of a category without an equality predicate on objects, and a dependent-type version of equality on objects (can only compare arrows if they are in the same hom-collection). 
A small category with an equality predicate on its objects admits a(n essentially) surjective functor from a discrete category if we assume enough choice. In ordinary foundations (such as ZF(C), NBG or variants), Vopěnka's principle is a large cardinal axiom equivalent to the assertion that there are no subcategories of a locally presentable category (e.g. $Set$) which are simultaneously large (have a class of objects) and discrete. The principle can be seen a shadow in ordinary foundations of the idea that there should be categories which are really just too big to have a collection of objects that behaves like a set. Notice that one can  form the posetal coreflection $Pos(C)$ of a category $C$ (it has the same objects and there is a unique arrow in $Pos(C)$ between any two objects if and only if there is any arrow between the analogous objects in $C$), and even take the core (the largest subgroupoid) of this. But we cannot get a category with an equality predicate on its objects unless we are happy to form some sort of quotient of $Core(Pos(C))$ to get a discrete category, and then it requires serious use of global choice on these super-large 'collections' to turn the canonical functor $Core(C)\to Core(Pos(C))/\sim$ into a functor $Core(Pos(C))/\sim \to Core(C) \to C$ to get an essentially surjective functor from a discrete category.
As a sort of half-way between this notion of category which is too large to be a class, we have the first-order characterisation of the category of classes, otherwise known as algebraic set theory. One could apply the more philosophical ideas from the above paragraphs to the very concrete definitions of algebraic set theory (see for instance section 3.1 of this introduction). One would then have a category of classes which is itself genuinely not a meta-class, nor some sort of collection which behaves like a class, nor just a 'class' corresponding to a large cardinal in model/universe of set theory containing the current model/universe. This would probably require playing around with the axiom (US) (section 3.1 here).
Mike Shulman has some good comments on similar (though less extreme) ideas in this answer.
If one complains that this is just a first step, and really we want a whole hierarchy  of notions of 'bigger than anything we can come up with so far', then Michael Makkai has considered foundations (a sort of type theory, called by him FOLDS) in which it is impossible to consider equality (as above), isomorphism (as might be considered natural in 2-categories, for example), equivalence,... so that we can really only talk about each of these notions if we are working in an $n$-category for some finite $n$, and in general the only available generalised notion of equivalence is full-blown $\omega$-equivalence of $\omega$-categories. But this sort of approach has not been thought of in the sense of making larger and larger hierarchies of objects. (It has come up in Voevodsky's univalent foundations, but only from a homotopy point of view.)