We study the asymptotic behavior of Masur-Veech volumes as the genus goes to infinity. We show the existence of a complete asymptotic expansion of these volumes that depends only on the genus and the number of singularities. The computation of the first term of this asymptotics expansion was a long standing problem. This problem was recently solved in by Aggarwal using purely combinatorial arguments, and then by D. Chen, M. Moeller, D. Zagier and the author using algebro-geometric insights. Our proof relies on a combination of both methods.