This is in reply to Mike Barr's revised question, requiring regularity, and Dan Leivant's reply. Unfortunately, regular beths with limit subscript are inaccessible, which Mike wanted to avoid. Even more unfortunately, I believe it is known to be consistent (relative to very large cardinals) that there are no cardinals of the sort Mike wants. Specifically, it is consistent that GCH fails everywhere, a result of Foreman and Woodin. In such a universe, the requirement "for all lambda < kappa, 2^lambda<=kappa " implies that kappa is a limit cardinal; if you also require kappa to be regular, then it is weakly inaccessible. I believe that the violations of GCH in the Foreman-Woodin model are mild enough so that such kappas must actually be strongly inaccessible, but I'm not certain about this. Incidentally, the fact that the Foreman-Woodin model requires large cardinals for its construction does not mean that this model has any inaccessibles. It does mean that there are cardinals that are inaccessible (and measurable, and much more) in certain submodels. Andreas Blass