  The little Higgs mechanism produces a light 100 GeV Higgs while raising the
natural cutoff from 1 TeV to 10 TeV. We attempt an iterative little Higgs
mechanism to produce multiple factors of 10 between the cutoff and the 100 GeV
Higgs mass in a perturbative theory. In the renormalizable sector of the
theory, all quantum corrections to the Higgs mass proportional to mass scales
greater than 1 TeV are absent -- this includes quadratically divergent,
log-divergent, and finite loops at all orders. However, even loops proportional
to scales just a factor of 10 above the Higgs (or any other scalar) mass come
with large numerical factors that reintroduce fine-tuning. Top loops, for
example, produce an expansion parameter of not 1/(4 pi) but 1/5. The geometric
increase in the number of fields at higher energies simply exacerbates this
problem. We build a complete two-stage model up to 100 TeV, show that direct
sensitivity of the electroweak scale to the cutoff is erased, and estimate the
tuning due to large numerical factors. We then discuss the possibility, in a
toy model with only scalar and gauge fields, of generating a tower of little
Higgs theories and show that the theory quickly becomes a large-N gauge theory
with ~ N fundamental scalars. We find evidence that at least this toy model
could successfully generate light scalars with an exponentially large cutoff in
the absence of supersymmetry or strong dynamics. The fine-tuning is not
completely eliminated, but evidence suggests that this result is model
dependent. We then speculate as to how one might marry a working tower of
fields of this type at high scales to a realistic theory at the weak scale.
