  The Chirikov standard map family is a one-parameter family of
volume-preserving maps exhibiting hyperbolicity on a `large' but noninvariant
subset of phase space. Based on this predominant hyperbolicity and numerical
experiments, it is anticipated that the standard map has positive metric
entropy for many parameter values. However, rigorous analysis is notoriously
difficult, and it remains an open question whether the standard map has
positive metric entropy for any parameter value. Here we study a problem of
intermediate difficulty: compositions of standard maps with increasing
parameter. When the coefficients increase to infinity at a sufficiently fast
polynomial rate, we obtain a Strong Law, Central Limit Theorem, and
quantitative mixing estimate for Holder observables. The methods used are not
specific to the standard map and apply to a class of compositions of
`prototypical' 2D maps with hyperbolicity on `most' of phase space.
