We study the exact (one-loop) effective potential of the littlest Higgs model
and determine the dependence of physical quantities, such as the vacuum
expectation value v_W and mass m_h of the Higgs boson, on the fundamental
parameters of the Lagrangian--masses, couplings of new states, the fundamental scale f of the sigma model, and the coefficients of operators quadratically sensitive to the cutoff of the theory. On the one hand, we show that it is possible to have the electroweak ground state and a relatively large cutoff \Lambda = 4\pi f with f in the 2 TeV range without requiring unnaturally small coefficients for quadratically divergent quantities, and with only moderate
cancellations between the contribution of different sectors to the effective
potential of the Higgs. On the other hand, this cannot be achieved while at the
same time keeping m_h close to its (bantamweight) current lower bound of 114.4
GeV. The natural expectation for m_h is O(f), mainly because of large
logarithmically divergent contributions to the effective potential of the
top-quark sector. Even a fine-tuning at the level of O(10^{-2}) in the
coefficients of the quadratic divergences is not enough to produce small
physical Higgs masses, and the natural expectation is in the 800 GeV range
(cruiserweight) for f \sim 2 TeV. We conclude that the littlest Higgs model is
a solution of the little hierarchy problem, in the sense that it stabilizes the
electroweak symmetry breaking scale to be a factor of 100 less than the cutoff
of the theory, but this requires a quite large physical mass for the Higgs, and
hence precision electroweak studies should be redone accordingly. We also study finite temperature corrections.