This probably isn't "well known", and you might think it's cheating. Nevertheless, it's a very concrete, maximally weak, example of an $\infty$-category that isn't an $(\infty, n)$-category for any $n < \infty$. It's simply the free $\infty$-category on one cell in each dimension.

For "free" to make sense, you have to use a definition of $\infty$-category for which that makes sense (e.g. any one in which $\infty$-categories are defined as algebras for some particular monad or operad). For Segal-type definitions, it's not clear that it does make sense.

I say "very concrete" because the cells and operations of this free $\infty$-category can be described in an explicit combinatorial way, much as the elements of a free group admit an explicit combinatorial description -- except that this, of course, is more complex.

This probably isn't "well known", and you might think it's cheating. Nevertheless, it's a very concrete, maximally weak, example of an $\infty$-category that isn't an $(\infty, n)$-category for any $n < \infty$. It's simply the free $\infty$-category on one cell in each dimension.

For "free" to make sense, you have to use an appropriate definition of $\infty$-category (e.g. any one in which $\infty$-categories are defined as algebras for some particular monad or operad). For Segal-type definitions, it's not clear that it does make sense.

I say "very concrete" because the cells and operations of this free $\infty$-category can be described in an explicit combinatorial way, much as the elements of a free group admit an explicit combinatorial description -- except that this, of course, is more complex.