  We provide a new, elementary proof of the multiplicative independence of
pairwise distinct $\mathrm{GL}_2^+(\mathbb{Q})$-translates of the modular
$j$-function, a result due originally to Pila and Tsimerman. We are thereby
able to generalise this result to a wider class of modular functions. We show
that this class includes a set comprising modular functions which arise
naturally as Borcherds lifts of certain weakly holomorphic modular forms. For
$f$ a modular function belonging to this class, we deduce, for each $n \geq 1$,
the finiteness of $n$-tuples of distinct $f$-special points that are
multiplicatively dependent and minimal for this property. This generalises a
theorem of Pila and Tsimerman on singular moduli. We then show how these
results relate to the Zilber--Pink conjecture for subvarieties of the mixed
Shimura variety $Y(1)^n \times \mathbb{G}_{\mathrm{m}}^n$ and prove some
special cases of this conjecture.
