Assume that $\omega<\kappa_1<\dotsb< \kappa_n$ are infinite cardinals such that for each $1\le i\le n$ there is a $\kappa_i$-complete, $\kappa_i^+$-saturated ideal $\mathcal I_i\subset \mathcal P(\kappa_i)$.  Can you obtain a ZFC models which contains  $n$-many measurable cardinals? The natural candidate is  $L[\mathcal I_1,\dotsc, \mathcal I_n]$.

It is well known that the answer is yes for $n=1$ (see Kunen: [Some applications of iterated ultrapowers in set theory](https://doi.org/10.1016/0003-4843(70)90013-6). Ann. Math. Logic 1 (1970), 179–227.)