  We investigate the surface critical behavior of two-dimensional multilayered
aperiodic Ising models in the extreme anisotropic limit. The system under
consideration is obtained by piling up two types of layers with respectively
$p$ and $q$ spin rows coupled via nearest neighbor interactions $\lambda r$ and
$\lambda$, where the succession of layers follows an aperiodic sequence. Far
away from the critical regime, the correlation length $\xi_\perp$ is smaller
than the first layer width and the system exhibits the usual behavior of an
ordinary surface transition. In the other limit, in the neighborhood of the
critical point, $\xi_\perp$ diverges and the fluctuations are sensitive to the
non-periodic structure of the system so that the critical behavior is governed
by a new fixed point. We determine the critical exponent associated to the
surface magnetization at the aperiodic critical point and show that the
expected crossover between the two regimes is well described by a scaling
function. From numerical calculations, the parallel correlation length
$\xi_\parallel$ is then found to behave with an anisotropy exponent $z$ which
depends on the aperiodic modulation and the layer widths.
