  We consider one-dimensional excited random walks (ERWs) with i.i.d. markovian
cookie stacks in the non-boundary recurrent regime. We prove that under
diffusive scaling such an ERW converges in the standard Skorokhod topology to a
multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of
the limiting process are given explicitly in terms of those of the cookie
markov chain at a single site. While our results extend the results of
Dolgopyat and Kosygina (2012, ERWs with boundedly many cookies per site) and
Kosygina and Peterson (2016, ERWs with periodic cookie stacks), the approach
taken is very different and involves coarse graining of both the ERW and the
random environment changed by the walk. Through a careful analysis of the
environment left by the walk after each ``mesoscopic'' step, we are able to
construct a coupling of the ERW at this ``mesoscopic'' scale with a suitable
discretization of the limiting BMPE. The analysis is based on generalized
Ray-Knight theorems for the directed edge local times of the ERW stopped at
certain stopping times and evolving in both the original random cookie
environment and (which is much more challenging) in the environment created by
the walk after each ``mesoscopic'' step.
