  Let $\mathbb{F}_q(T)$ be the field of rational functions in one variable over
a finite field. We introduce the notion of a totally $T$-adic function: one
that is algebraic over $\mathbb{F}_q(T)$ and whose minimal polynomial splits
completely over the completion $\mathbb{F}_q(\!(T)\!)$. We give two proofs that
the height of a nonconstant totally $T$-adic function is bounded away from
zero, each of which provides a sharp lower bound. We spend the majority of the
paper providing explicit constructions of totally $T$-adic functions of small
height (via arithmetic dynamics) and minimum height (via geometry and computer
search). We also execute a large computer search that proves certain kinds of
totally $T$-adic functions of minimum height over $\mathbb{F}_2(T)$ do not
exist. The problem of whether there exist infinitely many totally $T$-adic
functions of minimum positive height over $\mathbb{F}_q(T)$ remains open.
Finally, we consider analogues of these notions under additional integrality
hypotheses.
