  In the theory of harmonic Maass forms and mock modular forms, mock theta
functions are distinguished examples which arose from $q$-hypergeometric
examples of Ramanujan. Recently, there has been a body of work on higher depth
mock modular forms. Here, we introduce distinguished examples of these forms
which we call higher depth mock theta functions and develop $q$-hypergeometric
expressions for them. We provide three examples of mock theta functions of
depth two, each arising by multiplying a classical mock theta function with a
certain specialization of a universal mock theta function. In addition, we give
their modular completions, and relate each to a $q$-hypergeometric series.

Number Theory