Given a variety X in characteristic p>0, the e-th F-blowup of X is defined to be the universal birational flattening of the e-times iterated Frobenius push-forward F^e_(*O_X) of the structure sheaf. The stabilization of the list of indecomposable summands of F^e_*(O_X) as e varies implies the stabilization of the of the F-blowup sequence. We will discuss these two properties in the simplest non-trivial cases, emphasizing computations in concrete examples.