  Braiding is a geometric concept that manifests itself in a variety of
scientific contexts from biology to physics, and has been employed to classify
bulk band topology in topological materials. Topological edge states can also
form braiding structures, as demonstrated recently in a type of topological
insulators known as M\"obius insulators, whose topological edge states form two
braided bands exhibiting a M\"obius twist. While the formation of M\"obius
twist is inspiring, it belongs to the simple Abelian braid group
$\mathbb{B}_2$. The most fascinating features about topological braids rely on
the non-Abelianness in the higher-order braid group $\mathbb{B}_N$ ($N \geq
3$), which necessitates multiple edge bands, but so far it has not been
discussed. Here, based on the gauge enriched symmetry, we develop a scheme to
realize non-Abelian braiding of multiple topological edge bands. We propose
tight-binding models of topological insulators that are able to generate
topological edge states forming non-Abelian braiding structures. Experimental
demonstrations are conducted in two acoustic crystals, which carry three and
four braided acoustic edge bands, respectively. The observed braiding structure
can correspond to the topological winding in the complex eigenvalue space of
projective translation operator, akin to the previously established point-gap
winding topology in the bulk of the Hatano-Nelson model. Our work also
constitutes the realization of non-Abelian braiding topology on an actual
crystal platform, but not based on the "virtual" synthetic dimensions.
