Well, the $n$th layer is a standard category.
We can even omit layers above $n$, or say they all contain only the identity cells.
What you wrote rather fits for (weak) $n$-groupoids (such as the fundamental groupoids of a topological space), where all cells are invertible.
In general, two $(n-1)$-cells $\alpha$ and $\beta$ are equivalent ($\alpha\simeq\beta$) in an $n$-category, if there is an invertible $n$-cell $\varphi:\alpha\to\beta$, i.e. there's also a $\psi:\beta\to\alpha$ such that $\psi\varphi=1_\alpha$ and $\varphi\psi=1_\beta$.
If $\alpha, \beta$ are $(n-2)$-cells, then we require $\simeq$ in the above equations in place of $=$.
And so on..
To see a specific example, consider the bicategory of rings and bimodules (with bimodules ${}_AM_B$ as arrows $A\to B$, tensor product as composition, and bimodule morphisms as 2-cells).
Here two bimodules are equivalent iff they are isomorphic, and two rings are equivalent iff there are bimodules between them, which are inverses to each other w.r.t tensor product (which is called they are Morita-equivalent).
Associativity is required on each layer up to (equivalence on that layer).
In the above example it's just $(M\otimes N)\otimes P\cong M\otimes (N\otimes P) $. Note that these entities are indeed not identical, for the same reason as $(A\times B)\times C\ne A\times(B\times C)$ for sets, but there's a natural isomorphism between them.
However, this condition for associativity in itself proved to be not sufficient, and further coherence conditions had to be posed, see e.g. the definition of a bicategory.
In weak higher categories these coherence issues are highly nontrivial.
Best Answer
The is a model structure on simplicial sets whose fibrant objects are "quasicategories", the simplicial sets in which every inner horn has a filler. If the 0-simplices are thought of as objects of a category and the 1-simplices as morphisms, then the horn filler condition gives a sense in which a quasicategory admits composites of all arrows, with composition associative up to a homotopy, which is itself well defined up to a higher homotopy, which is...and so on. The latter is the primeval concept of an $(\infty,1)$-category, and quasicategories are the most used model.
In fact the first model category for $(\infty,1)$-categories that was introduced was that of Bergner, which is on the category of simplicially enriched categories. This models an $(\infty,1)$-category as a "category" with morphisms of all dimensions, all admitting strictly associative compositions. It is very far from obvious that everything we'd like to call an $(\infty,1)$-category is equivalent to something of this form, but in fact this is the case: there is a Quillen equivalence between the model categories of simplicial categories and of quasicategories, showing that every $(\infty,1)$-category (viewed as a quasicategory) can be viewed as a simplicially enriched category.
The sense in which a quasicategory $Q$ is "equivalent" to some simplically enriched category $\mathcal C$ is that the homotopy categories $\mathrm{Ho}(Q)$ and $\mathrm{Ho}(\mathcal C)$ are equivalent, but also, roughly speaking, that $Q$ and $\mathcal C$ have the same mapping spaces. One can't formalize this directly by asking that there be a map inducing such equivalences $Q\to \mathcal C$, as they don't live in the same category, which is the reason for the introduction of the Quillen equivalence between the model structures.