The language $FOL(=, \in , <, C)$ is mono-sorted first order predicate language with extra-logical primitives of equality (and its axioms), set membership, strict smaller than binary relation, and a two place function denoting "the count of an element in a set".
Well ordering: $<$ well orders the universe of elements.
Extensionality over sets (i.e., non-elements)
Flatness: $x \in y \to y \not \in z$
Counting: $x \in S \to C^S_x = lim^* \{ C^S_m | m \in S \land m < x\}$
Where $lim^* X$ is the upper strict limit on $X$, i.e. the ordinal immediately strictly larger than every element of $X$.
All elements in this theory are to be considered as "ordinals".
What is the expressive power of this language? I mean what is the limit to define ordinals in this language? for example one can define a regular ordinal $l$ as an element that is a limit ordinal, that is not the limit to a set $x$ of ordinals strictly smaller than it where $\lim^* \{C^x_m| m \in x\} < l$, that way we can define an inaccessible ordinal as a regular limit of regular ordinals. So this language has the expressive power to express inaccessibility.
So at the level of which ordinal in the known large cardinal tower, this language would start to fail in providing a unique expression of?