  False theta functions are functions that are closely related to classical
theta functions and mock theta functions. In this paper, we study their modular
properties at all ranks by forming modular completions analogous to modular
completions of indefinite theta functions of any signature and thereby develop
a structure parallel to the recently developed theory of higher depth mock
modular forms. We then demonstrate this theoretical base on a number of
examples up to depth three coming from characters of modules for the vertex
algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invariants of
$3$-manifolds associated with gauge group $\mathrm{SU}(3)$.
