  We generalize the theory of Contou-Carr\`ere symbols to higher dimensions. To
an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$
denotes a commutative algebra over a field $k$, we associate an element
$(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k
= A$, and earlier constructions for $n = 1$, by Contou-Carr\`ere, and $n = 2$
by Osipov--Zhu. Our definition is based on the notion of \emph{higher
commutators} for central extensions of groups by spectra, thereby extending the
approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. Following
Beilinson--Bloch--Esnault for the case $n=1$, we allow $A$ to be arbitrary, and
do not restrict to artinian $A$. Previous work of the authors on Tate objects
in exact categories, and the index map in algebraic $K$-theory is essential in
anchoring our approach to its predecessors. We also revisit categorical formal
completions, in the context of stable $\infty$-categories. Using these tools,
we describe the higher Contou-Carr\`ere symbol as a composition of boundary
maps in algebraic $K$-theory, and conclude the article by proving a version of
Parshin--Kato reciprocity for higher Contou-Carr\`ere symbols.
