  In the extreme high intensity regime of electromagnetic (EM) waves in plasma,
the acceleration process is found to be dominated by the ponderomotive
acceleration (PA). While the wakefields driven by the ponderomotive force of
the relativistic intensity EM waves are important, they may be overtaken by the
PA itself in the extreme high intensity regime when the dimensionless vector
potential $a_0$ of the EM waves far exceeds unity. The energy gain by this
regime (in 1D) is shown to be (approximately) proportional to $a_0^2$. Before
reaching this extreme regime, the coexistence of the PA and the wakefield
acceleration (WA) is observed where the wave structures driven by the
wakefields show the phenomenon of multiple and folded wave-breakings.
Investigated are various signatures of the acceleration processes such as the
dependence on the mass ratio for the energy gain as well as the energy spectral
features. The relevance to high energy cosmic ray acceleration and to the
relativistic laser acceleration is considered.
