We continue some work on L[Card] (the universe constructed from the predicate for the cardinals) to look at L[Reg] where Reg is the class of uncountable regular cardinals. The latter is also a model of a rich combinatorial structure being, as it turns out, a Magidor iteration of Prikry forcings (using recent work of Ben-Neria). But it is limited in size, in fact is a rather 'thin' model. We show, letting O^s = O^sword be the least iterable structure with a measure which concentrates on measurable cardinals:

Theorem (ZFC) (a) Let S be a set, or proper class, of regular cardinals, then O^s is not an element of L[S].

(b) This is best possible, in that no smaller mouse M can be substituted for O^s.

(c) L[S] is a model of: GCH, Square's, Diamonds, Morasses etc and has Ramsey cardinals, but no measurable cardinals.