This is far from a complete answer, but perhaps it will help make rigorous the idea of $(\infty,n)$ as $(\infty,1)$ enriched in $(\infty,n-1)$, and thereby give you another approach to $(\infty,\infty)$. Recently, Rune Haugseng defended his PhD thesis at MIT under Haynes Miller and did the necessary legwork to discuss enriched infinity categories. You can find his website here, and he's posted both his thesis and his research statement there. Rune's work should link up nicely with any model for $(\infty,n)$ category which you choose

This is far from a complete answer, but perhaps it will help make rigorous the idea of $(\infty,n)$ as $(\infty,1)$ enriched in $(\infty,n-1)$, and thereby give you another approach to $(\infty,\infty)$. Recently, Rune Haugseng defended his PhD thesis at MIT under Haynes Miller and did the necessary legwork to discuss enriched infinity categories. You can find his website here, and he's posted both his thesis and his research statement there. Rune's work should link up nicely with any model for $(\infty,n)$ category you choose.