  We introduce Lagrange Spectra of closed-invariant loci for the action of
SL(2,R) on the moduli space of translation surfaces, generalizing the classical
Lagrange Spectrum, and we analyze them with renormalization techniques. A
formula for the values in such spectra is established in terms of the
Rauzy-Veech induction and it is used to show that any invariant locus has
closed Lagrange spectrum and values corresponding to pseudo-Anosov elements are
dense. Moreover we show that Lagrange spectra of arithmetic Teichm\"uller discs
contain an Hall's ray, giving an explicit bound for it via a second formula
which uses the classical continued fraction algorithm. In addition, we show the
equivalence of several definitions of bounded Teichm\"uller geodesics and
bounded type interval exchange transformations and we prove quantitative
estimates on excursions to the boundary of moduli space in terms of norms of
positive matrices in the Rauzy-Veech induction.
