  We reveal a wealth of nonlinear and recoil effects in the interaction between
individual low-energy electrons and samples comprising a discrete number of
states. Adopting a quantum theoretical description of combined free-electron
and two-level systems, we find a maximum achievable excitation probability of
$100\%$, which requires specific conditions relating to the coupling strength
and the transition symmetry, as we illustrate through calculations for dipolar
and quadrupolar modes. Strong recoil effects are observed when the kinetic
energy of the probe lies close to the transition threshold, although the
associated probability remains independent of the electron wave function even
when fully accounting for nonlinear interactions with arbitrarily complex
multilevel samples. Our work reveals the potential of free electrons to control
localized excitations and delineates the boundaries of such control.
