  In this paper, we recursively construct explicit elements of provably high
order in finite fields. We do this using the recursive formulas developed by
Elkies to describe explicit modular towers. In particular, we give two explicit
constructions based on two examples of his formulas and demonstrate that the
resulting elements have high order. Between the two constructions, we are able
to generate high order elements in every characteristic. Despite the use of the
modular recursions of Elkies, our methods are quite elementary and require no
knowledge of modular curves. We compare our results to a recent result of
Voloch. In order to do this, we state and prove a slightly more refined version
of a special case of his result.
