The project "HighGG" (funded by Agence Nationale de la Recherche as a JCJC grant) aims at getting a better understanding of the geometric behavior of (random) surfaces in the high genus limit. The models that are considered include combinatorial maps, Hurwitz numbers, half-(translation) surfaces and hyperbolic surfaces. Besides using and developing purely geometric/combinatorial techniques, this project crucially requires/aims at a good understanding of the asymptotic behavior of certain quantities (mostly number of objects/volumes), sometimes as two (or more) parameters tend to infinity at the same time.