In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other results in this article imply that this is the minimal possible saturation, and such an ideal carries strong properties: Any generic ultrapower is well-founded and contains the same reals as the forcing extension.

Is there anything known about lower bounds on the consistency strength of normal, $\omega_{n+1}$-saturated ideals on $[\omega_n]^{<\omega_1}$ for $n>1$? For $n = 1$, this is known to be equiconsistent with a Woodin cardinal, by results of Shelah and recent work of Jensen and Steel. (Thanks, Andres!)

Woodin's axiom $(∗)$, bounded forcing axioms, and precipitous ideals on $\omega_1$. J. Symbolic Logic77(2012), no. 2, 475–498. There they explain how to prove the equiconsistency of a Woodin cardinals with a strong ideal on $\omega_1$. They explicitly make use of the Jensen-Steel approach (which, in turn, uses the technique ofstacking mice) and some new ideas. $\endgroup$ – Andrés E. Caicedo Mar 24 '16 at 15:25