I would like to gain some basic intuition about iterated ultrapowers. I am perfectly happy with accepting the construction and can see that it fits into a fundamental role in many places (for example, it seems to be the correct way to compare tiny structures in inner model theory via various coiteration lemmas), but I'm having trouble seeing what happens morally when we take iterated ultrapowers.
For example, suppose $\kappa$ is the least measurable and $U$ a normal measure on it. In the ultrapower, $j(\kappa)>\kappa$ is now the least measurable. If we keep on iterating through $\mathrm{Ord}$ we will eventually push all measurables out of existence. So in this sense the final model is smaller than the one we started with (this is apart from the fact that the ultrapowers are constructed internally to $V$ and are thus trivially smaller in a very boring sense).
But on the other hand, if $(L_\alpha,U)$ is a mouse with the top cardinal $\kappa$, the ultrapower must, by elementarity, be $L_{j(\kappa)^+}$, and after iterating through $\mathrm{Ord}$ we get to $L$. So these ultrapowers seem to grow.
What I'd ultimately like to know is whether, heuristically speaking, iterating ultrapowers "adds stuff" or "takes stuff away". What kind of thought process should I go through when looking at a problem to be able to say, "Ah, I will iterate this ultrapower and then everything will be great"?