I'm currently trying to learn about etale homotopy for schemes as introduced by Artin-Mazur. I know that by the Artin-Mazur comparision theorem, it is possible to compute the etale homotopy type of certain class of varieties as the profinite completion of the complex points. However, in most other cases for schemes, it seems quite cumbersome to calculate the étale homotopy type of a locally noetherian scheme say. Are there any explicit computations of the étale homotopy type that are particularly helpful for understanding the general theory? Or am I missing something here?
Sorry if my question is a bit vague.