Abstract: Hypergeometric multisums are sequences that appear in enumeration problems in combinatorics, algebraic geometry, and mathematical physics. Using the theory of G-functions, we will prove a general theorem concerning the existence of asymptotic expansions. In addition, we will construct a finite set of alegbraic numbers (associated directly to the summand) that conjecturally coincides with the set of exponential growth rates of the above sequences. Finally we will give a proof of our conjecture in some special cases, as well as supporting examples.