TITLE: Anti-large cardinal principles
QUESTION [22 upvotes]: I'm interested in axioms that prevent the existence of large cardinals. More precisely:
(Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \Psi$ for some garden-variety large cardinal axiom $\Psi$.
Probably the most famous are:
Variety 1 Axioms of constructibility (e.g. $V=L$, $V=L[x]$).
$V=L$ for example implies that there are no measurable cardinals in $V$. Often we don't need the full power of $L$ to get anti-large cardinal features though. For instance:
Variety 2 $\square$-principles 
often prevent the existence of large cardinals via their anti-reflection features. For example $\square_\kappa$ prevents the existence of cardinals that reflect stationary sets to $\alpha < \kappa$ (and hence rule out cardinals like supercompacts).
Trivially, we might look at axioms of the following form:
Variety 3 $\neg \Phi$ or $\neg Con(\Phi)$ for some large cardinal axiom $\Phi$.
These obviously and boringly prevents the existence of a large cardinal in the model. Another possible candidate (formalisable in $NBG + \Sigma^1_1$ comprehension) is:
Variety 4 Inner model hypotheses.
For example, the vanilla inner model hypothesis that any parameter-free first-order sentence true in an inner model of an outer model of $V$ is already true in an inner model of $V$. This prevents the existence of inaccessibles in $V$. Other variants rule out cardinals not consistent with $L$.
Final example (if we're being super liberal about what we allow to be revised):
Variety 5 Choice axioms.
For instance $AC$ prevents the existence of Reinhardt (and super-Reinhardt, Berkeley etc.) cardinals in $V$ (assuming that these are, in fact, consistent with $ZF$, which is pretty non-trivial).
My question:
Are there other kinds of axiom with anti-large-cardinal features? Especially non-trivial ones (i.e. not like Variety 3).

REPLY [5 votes]: The following is part of unpublished work by Toshimichi Usuba:
Definition. A cardinal $\kappa$ is hyper-huge if for every $\lambda > \kappa$ there is an elementary embedding
$$
j \colon V \to M
$$
into some inner model $M$ such that $\operatorname{crit}(j) = \kappa$, $\lambda < j(\kappa)$ and $^{j(\lambda)} M \subseteq M$.
Definition. An inner model $M$ of $V$ is a ground if there is some set forcing $\mathbb P \in M$ and some $\mathbb P$-generic filter $G$ over $M$ such that $M[G] = V$.
Theorem (Usuba). If there is a hyper-huge cardinal, then there are only set many grounds.
He also proved that $V$ has a $\subseteq$-minimal ground (which is known as the bedrock) iff there are only set many grounds. Thus the nonexistence of the bedrock is an anti-large cardinal axiom.
On the other hand, in Inner Model Theoretic Geology, Fuchs and Schindler provided an example for which the bedrock does not exist, namely: If $L[E]$ is a pure extender model without a strong cardinal satisfying some 
'nice inner model theoretic properties' (*) then the bedrock of $L[E]$ does not exist. (**)
(*) $L[E]$ is assumed to be tame, fully iterable in $V$, internally not fully iterable as guided by $\mathcal{P}$-constructions and the extender sequence $E$ and its canonical extensions by $\operatorname{Col}(\omega, \theta)$, for cutpoints $\theta$ of $L[E]$, are ordinal definable in their respective models.
(**) Please note that they provide a much more detailed analysis of the class of grounds for this model.