As you point out, there are "inclusion" functors $\mathrm{Cat}_n\to \mathrm{Cat}_{n+1}$.  These inclusion functors admit both left and right adjoints (in the sense of functors between $(\infty,1)$-categories).    

The right adjoint $r\colon \mathrm{Cat}_{n+1}\to \mathrm{Cat}_n$ is a kind of truncation functor which effectively removes all $(n+1)$-morphisms which are not $(n+1)$-equivalences. 

The left adjoint $\ell\colon \mathrm{Cat}_{n+1}\to \mathrm{Cat}_n$ is a different functor, which *inverts* all $(n+1)$-morphisms. 

This gives at least two different possible towers $\{\mathrm{Cat}_n\}$ of $(\infty,1)$-categories.  You can form the (homotopy) inverse limit of each in $(\infty,1)$-categories, and perhaps either of these could be considered a definition of $\infty$-category.  (Other people have thought about his more than I have, and may be able to tell you if these are really workable.)  The main thing to take here is that there is probably more than one decent choice.

The kind of phenomenon can also be seen in strict $\infty$-categories, where there are at least two different definitions of the notion of a $k$-morphism being a $k$-equivalence: an *inductive* definition, and a *coinductive* definition.  Which choice you make inevitably feeds into the choice of definiton of weak equivalence between strict $\infty$-cats.
For instance, Yves Lafont, Francois Metayer, Krzysztof Worytkiewicz, in "A folk model structure for omega cat", use a coinductive definition of $k$-equivalence.