Has anyone considered axioms to the effect that: "The axiom of constructibility fails very very badly?" If I'm not mistaken, the axiom of constructibility basically says that the universe has no (non-trivial) inner models. Has anyone considered axioms of the opposite flavour, basically asserting that the universe has many inner models?
Apologies if I've misunderstood the axiom of constructibility, it is very confusing to me.
 A: The axiom of constructibility is indeed equivalent to "there are no inner models", but this is not what it "essentially says". This would be sort of a backwards way of understanding the axiom.
Instead the axiom says that we can find a hierarchy where every set appears in some stage, and it is first-order definable over the previous stage in the hierarchy. In contrast, when we consider the von Neumann we take all the subsets of the previous step, definable or not.
It turns out that two models with the same ordinals have the same constructible universe. So in turn it follows that the axiom $V=L$ is equivalent to saying that there is no inner model (note, however, that "there are no inner models" is not a first-order statement).
If anything is the opposite of $V=L$ it would be the existence of $0^\#$. This axiom implies that not only $V\neq L$, but it is in fact very far away from $L$. What does that mean? If we start with $L$ and we add using forcing a real number, or even a proper class of new sets, then the resulting model will be somewhat close to $L$. In what sense do we mean close? If $X$ is any uncountable set of ordinals, then there is some $Y\in L$ which is a set of ordinals, $X\subseteq Y$ and $|X|=|Y|$. This is known as the covering lemma.
The existence of $0^\#$ is equivalent to saying that the covering lemma fails, and it turns out that it fails in many many places, when it fails. For example the set $\{\aleph_n\mid n\in\omega\}$ cannot be covered by any set of size $<\aleph_\omega$ which is in $L$. The reason is that $\aleph_\omega$ is regular in $L$, so any unbounded subset must have size $\aleph_\omega$.
If you want to extend this further, there is the notion of relative constructibility which says that there is some set $A$ from which the universe is constructible. And so we can define the notion of $A^\#$ in a similar fashion.
So the "ultimate anti-$L$ axiom" would be something of the form $\forall A(V\neq L[A])$. This axiom is a consequence of large cardinals such as strongly compact cardinals (and in fact much much weaker than that). And these axioms are used on a regular basis in set theory.
