  In this article we study an abelian analogue of Schanuel's conjecture. This
conjecture falls in the realm of the generalised period conjecture of Y.
Andr{\'e}. As shown by C. Bertolin, the generalised period conjecture includes
Schanuel's conjecture as a special case. Extending methods of Bertolin, it can
be shown that the abelian analogue of Schanuel's conjecture we consider, also
follows from Andr{\'e}'s conjecture. C. Cheng et al. showed that the classical
Schanuel's conjecture implies the algebraic independence of the values of the
iterated exponential function and the values of the iterated logarithmic
function, answering a question of M. Waldschmidt. We then investigate a similar
question in the setup of abelian varieties.
