Yes, Gauss investigated the AGM when he was very young. Also, later, it became an ingredient in an ultra-fast method of computing digits of pi.https://twitter.com/RickWicklin/status/1039951919207333888 …
Actually, the AGM originated with Lagrange, but Gauss arrived at it independently (when he was about fourteen/fifteen.) What Gauss did was to examine it as an analytic function of a complex variable -- which is what really reveals how utterly amazing a beast it is.