  Given a subfield $F$ of $\mathbb{C}$, we study the linear disjointess of the
field $E$ generated by iterated exponentials of elements of $\overline{F}$, and
the field $L$ generated by iterated logarithms, in the presence of Schanuel's
conjecture. We also obtain similar results replacing $\exp$ by the modular
$j$-function, under an appropriate version of Schanuel's conjecture, where
linear disjointness is replaced by a notion coming from the action of
$\mathrm{GL}_2$ on $\mathbb{C}$. We also show that for certain choices of $F$
we obtain unconditional versions of these statements.
