  Band structure for a crystal generally consists of connected components in
energy-momentum space, known as band complexes. Here, we explore a fundamental
aspect regarding the maximal number of bands that can be accommodated in a
single band complex. We show that in principle a band complex can have no
finite upper bound for certain space groups. It means infinitely many bands can
entangle together, forming a connected pattern stable against
symmetry-preserving perturbations. This is demonstrated by our developed
inductive construction procedure, through which a given band complex can always
be grown into a larger one by gluing a basic building block to it. As a
by-product, we demonstrate the existence of arbitrarily large accordion type
band structures containing $N_C=4n$ bands, with $n\in\mathbb{N}$.
