  We study the probability density function $P(h_m,L)$ of the maximum relative
height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite
size $L$. For all these lattice models, in the large $L$ limit, a central limit
argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a
universal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12}
w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airy
distribution function. For one instance of these models, corresponding to the
extremely anisotropic Ising model in two dimensions, this result is obtained by
an exact computation using transfer matrix technique, valid for any $L>0$.
These arguments and exact analytical calculations are supported by numerical
simulations, which show in addition that the subleading scaling function is
also universal, up to a non universal amplitude, and simply given by the
derivative of the Airy distribution function $f'(x)$.
