  We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on
$[0,1]$ with a polynomially increasing contraction rate. We look at subsets of
such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ
f_{i_3}\circ...$ if $i_n>\Phi(i_{n-1})$ for certain increasing functions
$\Phi:\mathbb{N}\rightarrow\mathbb{N}$. We compute both the Hausdorff and
packing dimensions of such sets. Our results generalize work of Ramharter which
shows that the set of continued fractions with strictly increasing digits has
Hausdorff dimension 1/2.
